Multiplexer, high-frequency front end circuit, and communication device

ABSTRACT

A multiplexer includes N acoustic wave filters each including one end connected in common and having a different pass band, in which when the N acoustic wave filters are in order from a side of a lower frequency of the pass band, at least one n-th acoustic wave filter among the N acoustic wave filters excluding an acoustic wave filter having the highest frequency of the pass band includes one or more acoustic wave resonators including a support substrate, a silicon nitride film laminated on the support substrate, a silicon oxide film laminated on the silicon nitride film, a piezoelectric body laminated on the silicon oxide film, and an IDT electrode provided on the piezoelectric body. All acoustic wave filters having a pass band in a higher frequency than a frequency of a pass band of the n-th acoustic wave filter satisfy f hl_t   (n) &gt;f u   (m)  or f hl_t   (n) &lt;f l   (m) .

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of priority to Japanese Patent Application No. 2017-154239 filed on Aug. 9, 2017 and is a Continuation Application of PCT Application No. PCT/JP2018/027358 filed on Jul. 20, 2018. The entire contents of each application are hereby incorporated herein by reference.

BACKGROUND OF THE INVENTION 1. Field of the Invention

The present invention relates to a multiplexer including two or more acoustic wave filters, and a high-frequency front end circuit and a communication device including the multiplexer.

2. Description of the Related Art

A multiplexer has been widely used in a high-frequency front end circuit of a cellular phone or a smartphone. For example, a multiplexer as a branching filter disclosed in Japanese Unexamined Patent Application Publication No. 2014-68123 has two or more band pass filters for different frequencies. Additionally, each of the band pass filters is defined by a surface acoustic wave filter chip. Each of the surface acoustic wave filter chips includes a plurality of surface acoustic wave resonators.

Japanese Unexamined Patent Application Publication No. 2010-187373 discloses an acoustic wave device formed by laminating an insulating film made of silicon dioxide and a piezoelectric substrate made of lithium tantalate on a support substrate made of silicon. Then, the support substrate and the insulating film are bonded to each other by a (111) plane of the silicon, thereby improving heat resistance.

In the multiplexer as disclosed in Japanese Unexamined Patent Application Publication No. 2014-68123, a plurality of acoustic wave filters for different frequencies are commonly connected to an antenna end side.

The inventors of preferred embodiments of the present application have discovered that, when a structure is included in which a piezoelectric body made of lithium tantalate is laminated directly or indirectly on a support substrate made of silicon, a plurality of higher-order modes appear on a higher frequency side than a main mode to be used. When such an acoustic wave resonator is used for an acoustic wave filter on a lower pass band side in a multiplexer, there is a risk that a ripple due to the higher-order mode of the acoustic wave filter will appear in a pass band of another acoustic wave filter on a higher pass band side in the multiplexer. That is, when the higher-order mode of the acoustic wave filter on the lower pass band side in the multiplexer is located within the pass band of the other acoustic wave filter on the higher pass band side, the ripple occurs in the pass band. Therefore, there is a risk that filter characteristics of the other acoustic wave filter will deteriorate.

SUMMARY OF THE INVENTION

Preferred embodiments of the present invention provide multiplexers in each of which a ripple due to a higher-order mode does not easily occur in another band pass filter, high-frequency front end circuits and communication devices including the multiplexers.

The inventors of preferred embodiments of the present application have discovered that, in an acoustic wave resonator in which a piezoelectric body made of lithium tantalate is laminated directly or indirectly on a support substrate made of silicon, first to third higher-order modes, which will be described later, appear on a higher frequency side than a main mode.

Multiplexers according to preferred embodiments of the present application each prevent at least one higher-order mode of first, second, and third higher-order modes from occurring in a pass band of another filter.

A multiplexer according to a preferred embodiment of the present invention includes N (where, N is an integer equal to or greater than 2) acoustic wave filters each including one end connected in common and having a different pass band, in which when the N acoustic wave filters are denoted as a first acoustic wave filter, a second acoustic wave filter (2), . . . , an N-th acoustic wave filter in order from a side of a lower frequency of the pass band, at least one n-th acoustic wave filter (1≤n<N) among the N acoustic wave filters excluding an acoustic wave filter having the highest frequency of the pass band includes one or more acoustic wave resonators, a t-th acoustic wave resonator among the one or more acoustic wave resonators includes a support substrate having Euler angles (φ_(Si), θ_(Si), ψ_(Si)) and made of silicon, a silicon nitride film laminated on the support substrate, a silicon oxide film laminated on the silicon nitride film, a piezoelectric body laminated on the silicon oxide film, having Euler angles (φ_(LT) in a range of about 0°± about 5°, θ_(LT), ψ_(LT) in a range of about 0°± about 15°), and made of lithium tantalate, and an IDT electrode provided on the piezoelectric body, when a wavelength determined by an electrode finger pitch of the IDT electrode is taken as λ in the t-th acoustic wave resonator, a thickness normalized by the wavelength λ is a wavelength normalized thickness, when setting values of T_(LT) as a wavelength normalized thickness of the piezoelectric body, θ_(LT) as an Euler angle of the piezoelectric body, T_(S) as a wavelength normalized thickness of the silicon oxide film, T_(N) as a wavelength normalized thickness of the silicon nitride film, T_(E) as a wavelength normalized thickness of the IDT electrode converted to a thickness of aluminum, obtained by a product of a value obtained by dividing a density of the IDT electrode by a density of aluminum and a wavelength normalized thickness of the IDT electrode, Nisi as a propagation orientation in the support substrate, and T_(Si) as a wavelength normalized thickness of the support substrate, at least one of frequencies f_(hs_t) ^((n)) of first, second, and third higher-order modes (where, s is 1, 2, or 3, a case that s is 1 indicates a frequency of the first higher-order mode, a case that s is 2 indicates a frequency of the second higher-order mode, and a case that s is 3 indicates a frequency of the third higher-order mode) determined by an expression (1) and an expression (2) indicated below determined by the T_(LT), the θ_(LT), the T_(Si) the T_(N), the T_(E), the ψ_(Si), and the T_(Si), and all m acoustic wave filters (n<m≤N) each having a pass band of a higher frequency than a frequency of a pass band of the n-th acoustic wave filter (n) satisfy an expression (3) or an expression (4) indicated below.

$\begin{matrix} {V_{\text{?}} = {{a_{T_{\text{?}}}^{(3)}\left( {\left( {T_{\text{?}} + c_{\text{?}}} \right)^{3} + b_{\text{?}}^{(3)}} \right)} + {{a_{T_{\text{?}}}^{(2)}\left( {\left( {T_{\text{?}} + c_{\text{?}}} \right)^{2} + b_{\text{?}}^{(2)}} \right)}{a_{\text{?}}^{(1)}\left( {T_{\text{?}} + c_{\text{?}}} \right)}} + {a_{\text{?}}^{(\text{?})}\left( {\left( {T_{S} + c_{\text{?}}} \right)^{2} + b_{\text{?}}^{(2)}} \right)} + {a_{T_{S}}^{(1)}\left( {T_{S} + c_{T_{\text{?}}}} \right)} + {a_{\text{?}}^{(\text{?})}\left( {\left( {T_{N} + c_{\text{?}}} \right)^{2} + b_{\text{?}}^{(2)}} \right)} + {a_{T_{\text{?}}}^{(1)}\left( {T_{N} + c_{T_{\text{?}}}} \right)} + {a_{T_{E}}^{(1)}\left( {T_{E} + c_{T_{E}}} \right)} + {a_{\psi_{\text{?}}}^{(5)}\left( {\left( {\psi_{Si} + c_{\psi_{\text{?}}}} \right)^{5} + b_{\text{?}}^{(5)}} \right)} + {a_{\psi_{\text{?}}}^{(4)}\left( {\left( {\psi_{Si} + c_{\psi_{Si}}} \right)^{4} + b_{\psi_{\text{?}}}^{(4)}} \right)} + {a_{\psi_{Si}}^{(3)}\left( {\left( {\psi_{Si} + c_{\psi_{Si}}} \right)^{3} + b_{\psi_{Si}}^{(3)}} \right)} + {a_{\psi_{Si}}^{(2)}\left( {\left( {\psi_{Si} + c_{\psi_{\text{?}}}} \right)^{2} + b_{\psi_{Si}}^{(2)}} \right)} + {a_{\psi_{Si}}^{(1)}\left( {\psi_{Si} + c_{\psi_{Si}}} \right)} + e}} & {{Expression}\mspace{14mu} (1)} \\ {\mspace{79mu} {{f_{h_{s}\_ \; t}^{(n)} = \frac{V_{h_{s}\_ \; t}}{\lambda_{t}^{(n)}}},\left( {{s = 1},2,3} \right)}} & {{Expression}\mspace{14mu} (2)} \\ {\mspace{79mu} {f_{{hs}\; \_ \; t}^{(n)} > f_{u}^{(m)}}} & {{Expression}\mspace{14mu} (3)} \\ {\mspace{79mu} {{f_{{hs}\; \_ \; t}^{(n)} < f_{l}^{(m)}}{\text{?}\text{indicates text missing or illegible when filed}}}} & {{Expression}\mspace{14mu} (4)} \end{matrix}$

The f_(hs_t) ^((n)) represents a frequency of a higher-order mode corresponding to the s in the t-th acoustic wave resonator included in the n-th acoustic wave filter, the λ_(t) ^((n)) is a wavelength determined by the electrode finger pitch of the IDT electrode in the t-th acoustic wave resonator included in the n-th acoustic wave filter, the f_(u) ^((m)) is a frequency of a high band side end portion of the pass band in each of the m-th acoustic wave filters, and the f_(l) ^((m)) is a frequency of a low band side end portion of the pass band in each of the m-th acoustic wave filters.

Note that each coefficient in the expression (1) is each value shown in Table 1, Table 2, or Table 3 indicated below for each value of the s and each crystal orientation of the support substrate.

TABLE 1 s = 1, First Higher-Order Mode Si(100) Si(110) Si(111) a_(TLT) ⁽³⁾ 0 0 0 a_(TLT) ⁽²⁾ 0 0 0 a_(TLT) ⁽¹⁾ −128.109974 −84.392576 −78.4352 b_(TLT) ⁽³⁾ 0 0 0 b_(TLT) ⁽²⁾ 0 0 0 c_(TLT) −0.2492038 −0.247604 −0.24838 a_(TS) ⁽²⁾ 0 0 0 a_(TS) ⁽¹⁾ −109.6889 −182.2936559 −485.867 b_(TS) ⁽²⁾ 0 0 0 c_(TS) −0.249363 −0.2498958 −0.24942 a_(TN) ⁽²⁾ −337.59528 −198.4171235 −264.804 a_(TN) ⁽¹⁾ −109.08389 38.137636 −20.3216 b_(TN) ⁽²⁾ −0.0262274 −0.04671597 −0.04389 c_(TN) −0.29617834 0.369166 −0.34988 a_(TE) ⁽¹⁾ 175.4682 13.0363945 0 c_(TE) −0.14826 −0.14979166 0 a_(ψSi) ⁽⁵⁾ 0 0 0 a_(ψSi) ⁽⁴⁾ 0 0.000489723 0.000503 a_(ψSi) ⁽³⁾ 0.0236358 −5.09E−05 0.006871 a_(ψSi) ⁽²⁾ −0.0357088 −1.017335189 −0.80395 a_(ψSi) ⁽¹⁾ −34.8157175 0 −5.57553 b_(ψSi) ⁽⁵⁾ 0 0 0 b_(ψSi) ⁽⁴⁾ 0 −2150682.513 −352545 b_(ψSi) ⁽³⁾ 0 −21460.18941 2095.948 b_(ψSi) ⁽²⁾ −288.415605 −970.8815104 −470.617 c_(ψSi) −22.5 −36.8125 −33.3025 e 5251.687898 5092.365583 4851.236

TABLE 2 s = 2, Second Higher-Order Mode Si(100) Si(110) Si(111) a_(TLT) ⁽³⁾ 0 0 0 a_(TLT) ⁽²⁾ 2285.602094 3496.38329 −2357.61 a_(TLT) ⁽¹⁾ −538.88053 −1081.86178 −1308.55 b_(TLT) ⁽³⁾ 0 0 0 b_(TLT) ⁽²⁾ −0.0016565 −0.001741462 −0.00166 c_(TLT) −0.251442 −0.2501547 −0.2497 a_(TS) ⁽²⁾ −3421.09725 −4927.3017 −3633.11 a_(TS) ⁽¹⁾ −1054.253 −992.33158 −1006.69 b_(TS) ⁽²⁾ −0.0016565 −0.2551083 −0.00166 c_(TS) −0.2514423 0.2551 −0.25019 a_(TN) ⁽²⁾ 1042.56084 −423.87007 −135.325 a_(TN) ⁽¹⁾ 159.11219 80.7948 −106.73 b_(TN) ⁽²⁾ −0.02613905 −0.05219411 −0.0486 c_(TN) −0.2961538 −0.36996 −0.39884 a_(TN) ⁽¹⁾ −171.153846 −637.391944 −585.696 c_(TE) 0.15 −0.151236 −0.14932 a_(ψSi) ⁽⁵⁾ 0 0 0 a_(ψSi) ⁽⁴⁾ 0 −0.00098215 −0.00016 a_(ψSi) ⁽³⁾ −0.0038938 −0.002109232 −0.00037 a_(ψSi) ⁽²⁾ −0.00306409 2.25463 0.224668 a_(ψSi) ⁽¹⁾ 2.8538478 23.6872514 1.243381 b_(ψSi) ⁽⁵⁾ 0 0 0 b_(ψSi) ⁽⁴⁾ 0 −2959279.229 −399785 b_(ψSi) ⁽³⁾ 234.60436 −21928.45828 5.712562 b_(ψSi) ⁽²⁾ −289.82063 −1407.041187 −535.077 c_(ψSi) 22.78846 −39.1640886 −29.9806 e 5282.98076 5338.606811 5411.395

TABLE 3 s = 3, Third Higher-Order Mode Si(100) Si(110) Si(111) a_(TLT) ⁽³⁾ 0 0 0 a_(TLT) ⁽²⁾ 0 0 3595.754 a_(TLT) ⁽¹⁾ −782.3425 −1001.237815 −592.246 b_(TLT) ⁽³⁾ 0 0 0 b_(TLT) ⁽²⁾ 0 0 −0.00164 c_(TLT) −0.254819 −0.2578947 −0.25367 a_(TS) ⁽²⁾ −14897.59116 0 0 a_(TS) ⁽¹⁾ −599.8312 −686.9212563 −438.155 b_(TS) ⁽²⁾ −0.00162005 0 0 c_(TS) −0.25682 −0.25546558 −0.25562 a_(TN) ⁽²⁾ 0 0 0 a_(TN) ⁽¹⁾ 0 125.557557 15.72663 b_(TN) ⁽²⁾ 0 0 0 c_(TN) 0 −0.349392713 −0.40872 a_(TE) ⁽¹⁾ −154.8823 −764.8758717 −290.54 c_(TE) −0.14819277 −0.15303643 −0.15149 a_(ψSi) ⁽⁵⁾ 0 0 0 a_(ψSi) ⁽⁴⁾ 0 0 −0.00073 a_(ψSi) ⁽³⁾ 0.010467682 −0.000286554 −0.00318 a_(ψSi) ⁽²⁾ −0.196913569 0.67197739 0.969126 a_(ψSi) ⁽¹⁾ 0.3019959 0.197549 0.359421 b_(ψSi) ⁽⁵⁾ 0 0 0 b_(ψSi) ⁽⁴⁾ 0 −0.000204665 0 b_(ψSi) ⁽³⁾ 0 −14837.92017 670.2052 b_(ψSi) ⁽²⁾ −240.3687037 −1590.306348 −525.572 c_(ψSi) 24.4578313 −41.9028 −31.1239 e 5730.906036 5574.008097 5675.837

A multiplexer according to a preferred embodiment of the present invention includes N (where, N is an integer equal to or greater than 2) acoustic wave filters each including one end connected in common and having a different pass band, in which when the N acoustic wave filters are denoted as a first acoustic wave filter, a second acoustic wave filter, . . . , an N-th acoustic wave filter in order from a side of a lower frequency of the pass band, at least one n-th acoustic wave filter (1≤n<N) among the N acoustic wave filters excluding an acoustic wave filter having the highest frequency of the pass band includes one or more acoustic wave resonators, a t-th acoustic wave resonator among the one or more acoustic wave resonators includes a support substrate having Euler angles (φ_(Si), θ_(Si), ψ_(Si)) and made of silicon, a silicon nitride film laminated on the support substrate, a silicon oxide film laminated on the silicon nitride film, a piezoelectric body laminated on the silicon oxide film, having Euler angles (φ_(LT) in a range of about 0°± about 5°, θ_(LT), ψ_(LT) in a range of about 0°± about 15°), and made of lithium tantalate, and an IDT electrode provided on the piezoelectric body, when a wavelength determined by an electrode finger pitch of the IDT electrode is λ in the t-th acoustic wave resonator, a thickness normalized by the wavelength λ is a wavelength normalized thickness, when setting values of T_(LT) as a wavelength normalized thickness of the piezoelectric body, θ_(LT) as an Euler angle of the piezoelectric body, T_(S) as a wavelength normalized thickness of the silicon oxide film, T_(N) as a wavelength normalized thickness of the silicon nitride film, T_(E) as a wavelength normalized thickness of the IDT electrode converted to a thickness of aluminum, obtained by a product of a value obtained by dividing a density of the IDT electrode by a density of aluminum and a wavelength normalized thickness of the IDT electrode, ψ_(Si) as a propagation orientation in the support substrate, and T_(Si) as a wavelength normalized thickness of the support substrate, at least one of frequencies f_(hs_t) ^((n)) of first, second, and third higher-order modes (where, s is 1, 2, or 3, a case that s is 1 indicates a frequency of the first higher-order mode, a case that s is 2 indicates a frequency of the second higher-order mode, and a case that s is 3 indicates a frequency of the third higher-order mode) determined by an expression (5) and an expression (2) indicated below determined by the T_(LT), the θ_(LT), the T_(Si) the T_(N), the T_(E), the ψ_(Si), and the T_(Si), and all m-th acoustic wave filters (n<m≤N) each having a pass band of a higher frequency than a frequency of a pass band of the acoustic wave filter (n) satisfy an expression (3) or an expression (4) indicated below.

$\begin{matrix} {V_{h} = {{a_{T_{LT}}^{(2)}\left( {\left( {T_{LT} - c_{T_{\text{?}}}} \right)^{2} - b_{T_{\text{?}}}^{(\text{?})}} \right)} + {a_{\text{?}}^{(1)}\left( {T_{LT} - c_{T_{\text{?}}}} \right)} + {a_{\text{?}}^{(2)}\left( {\left( {T_{S} - c_{T_{S}}} \right)^{2} - b_{T_{\text{?}}}^{(2)}} \right)} + {a_{T_{S}}^{(1)}\left( {T_{S} - c_{T_{\text{?}}}} \right)} + {a_{T_{N}}^{(3)}\left( {\left( {T_{N} - c_{T_{N}}} \right)^{3} - b_{T_{N}}^{(3)}} \right)} + {a_{T_{N}}^{(2)}\left( {\left( {T_{N} - c_{T_{N}}} \right)^{2} - b_{T_{N}}^{(2)}} \right)} + {a_{T_{N}}^{(1)}\left( {T_{N} - c_{T_{N}}} \right)} + {a_{T_{E}}^{(1)}\left( {T_{E} - c_{T_{E}}} \right)} + {a_{\psi_{Si}}^{(4)}\left( {\left( {\psi_{Si} - c_{\psi_{Si}}} \right)^{4} - b_{\psi_{Si}}^{(4)}} \right)} + {a_{\psi_{Si}}^{(3)}\left( {\left( {\psi_{Si} - c_{\psi_{Si}}} \right)^{3} - b_{\psi_{Si}}^{(3)}} \right)} + {a_{\psi_{Si}}^{(2)}\left( {\left( {\psi_{Si} - c_{\psi_{\text{?}}}} \right)^{2} - b_{\psi_{\text{?}}}^{(2)}} \right)} + {a_{\psi_{\text{?}}}^{(1)}\left( {\psi_{Si} - c_{\psi_{Si}}} \right)} + {a_{\theta_{LT}}^{(1)}\left( {\theta_{LT} - c_{\theta_{L\text{?}}}} \right)} + {{d_{T_{LT}T_{3}}\left( {T_{LT} - c_{T_{LT}}} \right)}\left( {T_{S} - {c_{T}}_{\text{?}}} \right)} + {{d_{\text{?}}\left( {T_{LT} - c_{LT}} \right)}\left( {T_{N} - c_{T_{N}}} \right)} + {{d_{\text{?}}\left( {T_{LT} - c_{T_{LT}}} \right)}\left( {\psi_{Si} - c_{\psi_{Si}}} \right)} + {{d_{T\text{?}}\left( {T_{S} - c_{T_{S}}} \right)}\left( {T_{N} - c_{T_{N}}} \right)} + {{d_{T\text{?}}\left( {T_{N} - c_{T_{N}}} \right)}\left( {\psi_{Si} - c_{\psi_{Si}}} \right)} + {{d_{T_{\text{?}}}\left( {T_{N} - c_{T_{N}}} \right)}\left( {\theta_{LT} - c_{\theta_{LT}}} \right)} + {{d_{T_{\text{?}}}\left( {T_{E} - c_{T_{E}}} \right)}\left( {\psi_{Si} - c_{\psi_{Si}}} \right)} + {{d_{\text{?}}\left( {\psi_{Si} - c_{\psi_{Si}}} \right)}\left( {\theta_{LT} - c_{\theta_{LT}}} \right)} + e}} & {{Expression}\mspace{14mu} (5)} \\ {\mspace{79mu} {{f_{h_{s}\_ \; t}^{(n)} = \frac{V_{h_{s}\_ \; t}}{\lambda_{t}^{(n)}}},\left( {{s = 1},2,3} \right)}} & {{Expression}\mspace{14mu} (2)} \\ {\mspace{79mu} {f_{{hs}\; \_ \; t}^{(n)} > f_{u}^{(m)}}} & {{Expression}\mspace{14mu} (3)} \\ {\mspace{79mu} {{f_{{hs}\; \_ \; t}^{(n)} < f_{l}^{(m)}}{\text{?}\text{indicates text missing or illegible when filed}}}} & {{Expression}\mspace{14mu} (3)} \end{matrix}$

The f_(hs_t) ^((n)) represents a frequency of a higher-order mode corresponding to the s in the t-th acoustic wave resonator included in the n-th acoustic wave filter, the λ_(t) ^((n)) is a wavelength determined by the electrode finger pitch of the IDT electrode in the t-th acoustic wave resonator included in the n-th acoustic wave filter, the f_(u) ^((m)) is a frequency of a high band side end portion of the pass band in each of the m-th acoustic wave filters, the f_(l) ^((m)) is a frequency of a low band side end portion of the pass band in each of the m-th acoustic wave filters, and each coefficient in the expression (5) is each value shown in Table 4, Table 5, or Table 6 indicated below for each value of the s and each crystal orientation of the support substrate.

TABLE 4 s = 1, First Higher- Order Mode Si(100) Si(110) Si(111) a_(TLT) ⁽²⁾ 0 0 0 a_(TLT) ⁽¹⁾ 0 0 0 b_(TLT) ⁽²⁾ 0 0 0 c_(TLT) 0 0 0 a_(TS) ⁽²⁾ 0 0 0 a_(TS) ⁽¹⁾ 0 0 534.5188318 b_(TS) ⁽²⁾ 0 0 0 c_(TS) 0 0 0.249010293 a_(TN) ⁽³⁾ 0 0 0 a_(TN) ⁽²⁾ 0 0 0 a_(TN) ⁽¹⁾ 0 0 −36.51741324 b_(TN) ⁽³⁾ 0 0 0 b_(TN) ⁽²⁾ 0 0 0 c_(TN) 0 0 0.35114806 a_(TE) ⁽¹⁾ 0 0 0 c_(TE) 0 0 0 a_(ψSi) ⁽⁴⁾ 0 0 0.000484609 a_(ψSi) ⁽³⁾ 0.022075968 0 0.005818261 a_(ψSi) ⁽²⁾ −0.18782287 0.081701713 −0.805302371 a_(ψSi) ⁽¹⁾ −33.85785847 10.57201342 −4.785681077 b_(ψSi) ⁽⁴⁾ 0 0 351437.8188 b_(ψSi) ⁽³⁾ 806.2400011 0 −1862.605341 b_(ψSi) ⁽²⁾ 270.2635345 986.4812738 471.945355 c_(ψSi) 20.26171875 37.73795535 32.87410926 a_(θLT) ⁽¹⁾ 0 0 0 c_(θLT) 0 0 0 d_(TLTTS) 0 0 0 d_(TLTTN) 0 0 0 d_(TLTψSi) 0 0 0 d_(TSTN) 0 0 1862.994192 d_(TNψSi) 0 0 0 d_(TNθLT) 0 0 0 d_(TEψSi) 0 0 0 d_(ψSiθLT) 0 0 0 e 5317.859375 5103.813161 4853.204861

TABLE 5 s = 2, Second Higher- Order Mode Si(100) Si(110) Si(111) a_(TLT) ⁽²⁾ 0 0 0 a_(TLT) ⁽¹⁾ −608.2898721 −1003.471473 −1270.018362 b_(TLT) ⁽²⁾ 0 0 0 c_(TLT) 0.25 0.253954306 0.249121666 a_(TS) ⁽²⁾ 0 0 0 a_(TS) ⁽¹⁾ −1140.654415 −1030.75867 −1039.830158 b_(TS) ⁽²⁾ 0 0 0 c_(TS) 0.249966079 0.255272408 0.250032531 a_(TN) ⁽³⁾ −3219.596725 0 2822.963403 a_(TN) ⁽²⁾ 555.8662451 0 −264.9680504 a_(TN) ⁽¹⁾ 465.8636149 53.04201209 −288.9461645 b_(TN) ⁽³⁾ 0.001081155 0 0.000392787 b_(TN) ⁽²⁾ 0.04654949 0 0.48934743 c_(TN) 0.378426052 0.376449912 0.392908263 a_(TE) ⁽¹⁾ 0 −622.7635558 −614.5885324 c_(TE) 0 0.151274165 0.15815224 a_(ψSi) ⁽⁴⁾ 0 −0.00096736 −0.000227305 a_(ψSi) ⁽³⁾ 0 −0.006772454 −0.000220017 a_(ψSi) ⁽²⁾ 0 2.203099663 0.31727324 a_(ψSi) ⁽¹⁾ 0.833288758 28.15768206 0.648998523 b_(ψSi) ⁽⁴⁾ 0 2959964.533 396965.3474 b_(ψSi) ⁽³⁾ 0 19143.61126 87.44425969 b_(ψSi) ⁽²⁾ 0 1447.367657 532.0008856 c_(ψSi) 22.51017639 40.50966608 29.90240729 a_(θLT) ⁽¹⁾ −1.501270796 −2.076046604 −2.376979261 c_(θLT) −52.08683853 −50.82249561 −49.04879636 d_(TLTTS) 0 0 0 d_(TLTTN) 0 0 0 d_(TLTψSi) 0 16.61849238 0 d_(TSTN) 0 1820.795615 1482.11565 d_(TNψSi) 0 3.625908485 −3.131543418 d_(TNθLT) 0 0 1607.412093 d_(TEψSi) 0 0 0 d_(ψSiθLT) 0 0 0.089566113 e 5326.187246 5356.110093 5418.323508

TABLE 6 s = 3, Third Higher- Order Mode Si(100) Si(110) Si(111) a_(TLT) ⁽²⁾ −14710.45271 0 0 a_(TLT) ⁽¹⁾ −764.4056124 −942.2882121 −582.1313356 b_(TLT) ⁽²⁾ 0.001558682 0 0 c_(TLT) 0.257243963 0.255649419 0.251712062 a_(TS) ⁽²⁾ −21048.18754 0 0 a_(TS) ⁽¹⁾ −508.6730943 −705.5211128 −400.0368899 b_(TS) ⁽²⁾ 0.001583662 0 0 c_(TS) 0.257243963 0.254751848 0.254357977 a_(TN) ⁽³⁾ 0 0 0 a_(TN) ⁽²⁾ 0 0 0 a_(TN) ⁽¹⁾ 0 97.59462013 24.94240828 b_(TN) ⁽³⁾ 0 0 0 b_(TN) ⁽²⁾ 0 0 0 c_(TN) 0 0.367793031 0.404280156 a_(TE) ⁽¹⁾ −276.7311066 −747.0884117 0 c_(TE) 0.1494796 0.152164731 0 a_(ψSi) ⁽⁴⁾ 0 0 −0.00075146 a_(ψSi) ⁽³⁾ 0.011363183 0.003532214 −0.002666357 a_(ψSi) ⁽²⁾ −0.23320473 0.218669312 1.006728665 a_(ψSi) ⁽¹⁾ 0.214067146 −11.24365221 0.523191515 b_(ψSi) ⁽⁴⁾ 0 0 381500.5075 b_(ψSi) ⁽³⁾ 180.0564368 20914.04622 −493.6094588 b_(ψSi) ⁽²⁾ 257.0498426 1548.182277 530.6814032 c_(ψSi) 22.31890092 39.72544879 30.82490272 a_(θLT) ⁽¹⁾ 0 0 −1.551626054 c_(θLT) 0 0 −49.16731518 d_(TLTTS) −13796.64706 0 1575.283126 d_(TLTTN) 0 0 0 d_(TLTψSi) 30.35701585 0 0 d_(TSTN) 0 0 0 d_(TNψSi) 0 0 0 d_(TNθLT) 0 0 0 d_(TEψSi) 28.27908094 0 0 d_(ψSiθLT) 0 0 −0.086544362 e 5700.075407 5563.854277 5688.418884

In a multiplexer according to a preferred embodiment of the present invention, the values of the T_(LT), the θ_(LT), the T_(Si) the T_(N), the T_(E), the ψ_(Si), and the T_(Si), are selected such that the frequencies f_(hs_t) ^((n)) of the first and second higher-order modes satisfy the expression (3) or the expression (4).

In a multiplexer according to a preferred embodiment of the present invention, the values of the T_(LT), the θ_(LT), the T_(Si) the T_(N), the T_(E), the ψ_(Si), and the T_(Si), are selected such that the frequencies f_(hs_t) ^((n)) of the first and third higher-order modes satisfy the expression (3) or the expression (4).

In a multiplexer according to a preferred embodiment of the present invention, the values of the T_(LT), the θ_(LT), the T_(Si) the T_(N), the T_(E), the ψ_(Si), and the T_(Si), are selected such that the frequencies f_(hs_t) ^((n)) of the second and third higher-order modes satisfy the expression (3) or the expression (4).

In a multiplexer according to a preferred embodiment of the present invention, the values of the T_(LT), the θ_(LT), the T_(Si) the T_(N), the T_(E), the ψ_(Si), and the T_(Si), are selected such that all the frequencies f_(hs_t) ^((n)) of the first, second, and third higher-order modes satisfy the expression (3) or the expression (4). In this case, ripples caused by respective responses by the first higher-order mode, the second higher-order mode, and the third higher-order mode do not appear in the pass band of the other acoustic wave filter.

In a multiplexer according to a preferred embodiment of the present invention, the wavelength normalized thickness T_(Si) of the support substrate satisfies T_(Si)> about 4.

In a multiplexer according to a preferred embodiment of the present invention, T_(Si)> about 10 is satisfied.

In a multiplexer according to a preferred embodiment of the present invention, T_(Si)> about 20 is satisfied.

In a multiplexer according to a preferred embodiment of the present invention, the wavelength normalized thickness of the piezoelectric body is equal to or less than about 3.5λ.

In a multiplexer according to a preferred embodiment of the present invention, the wavelength normalized thickness of the piezoelectric body is equal to or less than about 2.5λ.

In a multiplexer according to a preferred embodiment of the present invention, the wavelength normalized thickness of the piezoelectric body is equal to or less than about 1.5λ.

In a multiplexer according to a preferred embodiment of the present invention, the wavelength normalized thickness of the piezoelectric body is equal to or less than about 0.5λ.

In a multiplexer according to a preferred embodiment of the present invention, an antenna terminal to which one ends of the plurality of acoustic wave filters are connected in common is further included, and the acoustic wave resonator satisfying the expression (3) or the expression (4) is an acoustic wave resonator which is closest to the antenna terminal. In this case, ripples caused by the first, second, and third higher-order modes are more unlikely to occur in a pass band of the other acoustic wave filter.

In a multiplexer according to a preferred embodiment of the present invention, all of the one or more acoustic wave resonators are each the acoustic wave resonator satisfying the expression (3) or the expression (4). In this case, the ripple caused by at least one higher-order mode among the first, second, and third higher-order modes in the other acoustic wave filter is able to be more effectively reduced or prevented.

A multiplexer according to a preferred embodiment of the present invention may also be a duplexer.

Furthermore, a multiplexer according to a preferred embodiment of the present invention may further include an antenna terminal to which one ends of the plurality of acoustic wave filters are connected in common, and may be a composite filter in which three or more acoustic wave filters are connected in common on a side of the antenna terminal.

In a multiplexer according to a preferred embodiment of the present invention, the multiplexer is a composite filter device for carrier aggregation (CA).

Preferably, the acoustic wave filter including the one or more acoustic wave resonators in a multiplexer according to a preferred embodiment of the present invention is a ladder filter including a plurality of serial arm resonators and a plurality of parallel arm resonators. In this case, the influence of the higher-order mode is able to be more effectively reduced or prevented.

A high-frequency front end circuit according to a preferred embodiment of the present invention includes a multiplexer according to a preferred embodiment of the present invention, and a power amplifier.

A communication device according to a preferred embodiment of the present invention includes a high-frequency front end circuit including a multiplexer according to a preferred embodiment of the present invention and a power amplifier, and an RF signal processing circuit.

According to preferred embodiments of the present invention, at least one higher-order mode among a plurality of higher-order modes generated by at least one acoustic wave resonator of an acoustic wave filter on a lower pass band side does not easily occur within a pass band of another acoustic wave filter on a higher pass band side. Accordingly, deterioration in filter characteristics of the other acoustic wave filter does not easily occur. Therefore, it is possible to provide high-frequency front end circuits and communication devices each including a multiplexer having excellent filter characteristics.

The above and other elements, features, steps, characteristics and advantages of the present invention will become more apparent from the following detailed description of the preferred embodiments with reference to the attached drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a circuit diagram of a multiplexer according to a first preferred embodiment of the present invention.

FIG. 2 is a circuit diagram illustrating a first acoustic wave filter used in the multiplexer according to the first preferred embodiment of the present invention.

FIG. 3A is a schematic elevational cross-sectional view of an acoustic wave resonator used in the multiplexer according to the first preferred embodiment of the present invention, and FIG. 3B is a schematic plan view illustrating an electrode structure of the acoustic wave resonator.

FIG. 4 is a schematic diagram illustrating pass bands of first to fourth acoustic wave filters according to the first preferred embodiment of the present invention.

FIG. 5 is a diagram illustrating admittance characteristics of an acoustic wave resonator.

FIG. 6 is a diagram illustrating a relationship between a propagation orientation ψ_(Si) of a support substrate made of silicon and acoustic velocities of a main mode and a first higher-order mode.

FIG. 7 is a diagram illustrating a relationship between a wavelength normalized thickness of a piezoelectric body made of lithium tantalate and the acoustic velocities of the main mode and the first higher-order mode.

FIG. 8 is a diagram illustrating a relationship between a cut-angle (about 90°+θ_(LT)) of the piezoelectric body made of lithium tantalate and the acoustic velocities of the main mode and the first higher-order mode.

FIG. 9 is a diagram illustrating a relationship between a wavelength normalized thickness of a silicon oxide film and the acoustic velocities of the main mode and the first higher-order mode.

FIG. 10 is a diagram illustrating a relationship between a wavelength normalized thickness of a silicon nitride film and the acoustic velocities of the main mode and the first higher-order mode.

FIG. 11 is a diagram illustrating a relationship between a wavelength normalized thickness of an interdigital transducer (IDT) electrode and the acoustic velocities of the main mode and the first higher-order mode.

FIG. 12A is a diagram illustrating filter characteristics of a multiplexer of a comparative example, and FIG. 12B is a diagram illustrating filter characteristics of the multiplexer according to the first preferred embodiment of the present invention.

FIG. 13 is a diagram illustrating a relationship between a wavelength normalized thickness of the support substrate made of silicon and the phase maximum values of the first higher-order mode, a second higher-order mode, and a third higher-order mode.

FIG. 14 is a diagram illustrating a relationship between the propagation orientation ψ_(Si) of the support substrate made of silicon and acoustic velocities of the main mode and the second higher-order mode.

FIG. 15 is a diagram illustrating a relationship between the wavelength normalized thickness of the piezoelectric body made of lithium tantalate and the acoustic velocities of the main mode and the second higher-order mode.

FIG. 16 is a diagram illustrating a relationship between the cut-angle (about 90°+θ_(LT)) of the piezoelectric body made of lithium tantalate and the acoustic velocities of the main mode and the second higher-order mode.

FIG. 17 is a diagram illustrating a relationship between the wavelength normalized thickness of the silicon oxide film and the acoustic velocities of the main mode and the second higher-order mode.

FIG. 18 is a diagram illustrating a relationship between the wavelength normalized thickness of the silicon nitride film and the acoustic velocities of the main mode and the second higher-order mode.

FIG. 19 is a diagram illustrating a relationship between the wavelength normalized thickness of the IDT electrode and the acoustic velocities of the main mode and the second higher-order mode.

FIG. 20 is a diagram illustrating a relationship between the propagation orientation ψ_(Si) of the support substrate made of silicon and acoustic velocities of the main mode and the third higher-order mode.

FIG. 21 is a diagram illustrating a relationship between the wavelength normalized thickness of the piezoelectric body made of lithium tantalate and the acoustic velocities of the main mode and the third higher-order mode.

FIG. 22 is a diagram illustrating a relationship between the cut-angle (about 90°+θ_(LT)) of the piezoelectric body made of lithium tantalate and the acoustic velocities of the main mode and the third higher-order mode.

FIG. 23 is a diagram illustrating a relationship between the wavelength normalized thickness of the silicon oxide film and the acoustic velocities of the main mode and the third higher-order mode.

FIG. 24 is a diagram illustrating a relationship between the wavelength normalized thickness of the silicon nitride film and the acoustic velocities of the main mode and the third higher-order mode.

FIG. 25 is a diagram illustrating a relationship between the wavelength normalized thickness of the IDT electrode and the acoustic velocities of the main mode and the third higher-order mode.

FIG. 26 is a diagram illustrating a relationship between a film thickness of an LiTaO₃ film and a Q value in the acoustic wave device.

FIG. 27 is a diagram illustrating a relationship between the film thickness of the LiTaO₃ film and a temperature coefficient of frequency TCF in the acoustic wave device.

FIG. 28 is a diagram illustrating a relationship between the film thickness of the LiTaO₃ film and an acoustic velocity in the acoustic wave device.

FIG. 29 is a diagram illustrating a relationship between a thickness of the piezoelectric body made of LiTaO₃ and a band width ratio.

FIG. 30 is a diagram illustrating a relationship among a film thickness of an SiO₂ film, a material of a high acoustic velocity film, and the acoustic velocity.

FIG. 31 is a diagram illustrating a relationship among the film thickness of the SiO₂ film, an electromechanical coupling coefficient, and the material of the high acoustic velocity film.

FIG. 32 is a schematic configuration diagram of a communication device having a high-frequency front end circuit which is a preferred embodiment of the present invention.

FIG. 33 is a schematic diagram for explaining a crystal orientation.

FIG. 34 is a schematic diagram for explaining the crystal orientation.

FIG. 35 is a schematic diagram for explaining the crystal orientation.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The present invention will be made clear hereinafter through description of specific preferred embodiments of the present invention with reference to the drawings.

Each of the preferred embodiments described in the present specification is exemplary and a partial replacement or combination of configurations is also possible among different preferred embodiments.

First Preferred Embodiment

FIG. 1 is a circuit diagram of a multiplexer according to a first preferred embodiment of the present invention. A multiplexer 1 includes an antenna terminal 2. The antenna terminal 2 is, for example, a terminal connected to an antenna of a smartphone.

In the multiplexer 1, first to fourth acoustic wave filters 3 to 6 are connected in common to the antenna terminal 2. Each of the first to fourth acoustic wave filters 3 to 6 is a band pass filter.

FIG. 4 is a schematic diagram illustrating pass bands of the first to fourth acoustic wave filters 3 to 6. As illustrated in FIG. 4, the pass bands of the first to fourth acoustic wave filters are different. The pass bands of the first to fourth acoustic wave filters are referred to as first to fourth pass bands, respectively.

For frequency positions, the first pass band < the second pass band < the third pass band < the fourth pass band is satisfied. In the second to fourth pass bands, a low band side end portion is denoted by f₁ ^((m)), and a high band side end portion is denoted by f_(u) ^((m)). Note that the low band side end portion is a low band side end portion of the pass band. Additionally, the high band side end portion is a high band side end portion of the pass band. As the low band side end portion and the high band side end portion of the pass band, for example, an end portion of a frequency band of each band standardized by 3GPP or the like may preferably be used. Note that the multiplexer 1 according to the present preferred embodiment can be used as, for example, a composite filter device for carrier aggregation.

Here, (m) is 2, 3, or 4 according to the second to fourth pass bands.

The first to fourth acoustic wave filters 3 to 6 each include a plurality of acoustic wave resonators. FIG. 2 is a circuit diagram of the first acoustic wave filter 3. The first acoustic wave filter 3 includes serial arm resonators S1 to S3 and parallel arm resonators P1 and P2 each of which is configured of an acoustic wave resonator. That is, the first acoustic wave filter 3 is preferably a ladder filter. However, the number of serial arm resonators and the number of parallel arm resonators in the ladder filter are not limited to this.

Additionally, the second to fourth acoustic wave filters 4 to 6 are similarly configured as ladder filters in the present preferred embodiment, and include a plurality of serial arm resonators and a plurality of parallel arm resonators.

Note that the first to fourth acoustic wave filters 3 to 6 may have a circuit configuration other than the ladder filter as long as a plurality of acoustic wave resonators are included. For example, an acoustic wave filter in which an acoustic wave resonator is connected in series to a longitudinally coupled resonator acoustic wave filter may also be used. Furthermore, an acoustic wave filter in which a ladder filter is connected to a longitudinally coupled resonator acoustic wave filter may also be used. It is sufficient that the first acoustic wave filter 3 includes one or more acoustic wave resonators.

FIG. 3A is a schematic elevational cross-sectional view of an acoustic wave resonator of the serial arm resonators S1 to S3 or the parallel arm resonators P1 and P2 of the first acoustic wave filter 3, and FIG. 3B is a schematic plan view illustrating an electrode structure thereof.

An acoustic wave resonator 11 includes a support substrate 12, a silicon nitride film 13 laminated on the support substrate 12, a silicon oxide film 14, and a piezoelectric body 15 laminated on the silicon oxide film 14. That is, the silicon oxide film 14 is laminated between the silicon nitride film 13 and the piezoelectric body 15. Note that when the support substrate 12, the silicon nitride film 13, the silicon oxide film 14, and the piezoelectric body 15 are used as respective layers of the acoustic wave resonator 11, another layer may be laminated between the respective layers. The silicon oxide film 14 may include a plurality of layers, and may have a multilayer structure including an intermediate layer made of titanium, nickel, or the like, for example, between the plurality of layers. That is, a multilayer structure in which a first silicon oxide film, an intermediate layer, and a second silicon oxide film are laminated in this order from the support substrate 12 side may be used. A wavelength normalized thickness of the silicon oxide film 14 in this case indicates a thickness of the multilayer structure as a whole. In the same or similar manner, the silicon nitride film 13 may include a plurality of layers, and may have a multilayer structure including an intermediate layer made of titanium, nickel, or the like, for example, between the plurality of layers. That is, a multilayer structure in which a first silicon nitride film, an intermediate layer, and a second silicon nitride film 13 are laminated in this order from the support substrate 12 side may be used. A wavelength normalized thickness of the silicon nitride film 13 in this case indicates a thickness of the multilayer structure as a whole.

The support substrate 12 is made of silicon. The support substrate 12 is preferably made of single crystal silicon, for example, but does not have to be made of a complete single crystal, and it is sufficient that a crystal orientation is included. The silicon nitride film 13 is preferably an SiN film, for example. However, the SiN may be doped with other elements. The silicon oxide film 14 is preferably an SiO₂ film, for example. As long as the silicon oxide film 14 is made of silicon oxide, for example, a material obtained by doping SiO₂ with fluorine or the like may be included. The piezoelectric body 15 is made of lithium tantalate. The piezoelectric body 15 is preferably made of single crystal lithium tantalate, for example, but does not have to be made of a complete single crystal, and it is sufficient that a crystal orientation is included. In addition, as long as the piezoelectric body 15 is made of lithium tantalate, a material other than LiTaO₃ may be used. The lithium tantalate may be doped with Fe or the like.

Note that a thickness of the silicon oxide film 14 may be 0. In other words, the silicon oxide film 14 may not be provided.

An IDT (Interdigital Transducer) electrode 16 is provided on an upper surface of the piezoelectric body 15. More specifically, reflectors 17 a and 17 b are provided on both sides in an acoustic wave propagation direction of the IDT electrode 16, thus defining a single-port surface acoustic wave resonator. In the present preferred embodiment, the IDT electrode 16 is directly provided on the piezoelectric body 15, but the IDT electrode 16 may be indirectly provided on the piezoelectric body 15.

The inventors of preferred embodiments of the present application have discovered that, in the acoustic wave filter device in which the piezoelectric body 15 made of lithium tantalate is laminated directly or indirectly on the support substrate 12, when an acoustic wave is excited, responses of a plurality of higher-order modes appear on a higher frequency side than a main mode, other than a response of a main mode to be used. The plurality of higher-order modes will be described with reference to FIG. 5.

FIG. 5 is a diagram illustrating admittance characteristics of an example of an acoustic wave resonator in which a silicon nitride film, a silicon oxide film, and a piezoelectric body are laminated on a support substrate. As is apparent from FIG. 5, responses of first to third higher-order modes appear at frequency positions higher than a response of a main mode appearing near 3.9 GHz. The response of the first higher-order mode appears near 4.7 GHz, as indicated by the arrow. The response of the second higher-order mode appears near 5.2 GHz. The response of the third higher-order mode appears near 5.7 GHz. That is, when the frequency of the response of the first higher-order mode is denoted as f1, the frequency of the response of the second higher-order mode is denoted as f2, and the frequency of the response of the third higher-order mode is denoted as f3, f1<f2<f3 is established. Note that the frequency of the response of the higher-order mode is a peak position of impedance phase characteristics of the higher-order mode.

As described above, in the multiplexer in which the plurality of acoustic wave filters for different frequencies are commonly connected on the antenna terminal side, when the higher-order mode by the acoustic wave filter on the lower pass band side appears in the pass band of another acoustic wave filter on the higher pass band side in the multiplexer, a ripple is caused. Therefore, it is preferable that at least one higher-order mode among the first higher-order mode, the second higher-order mode, and the third higher-order mode does not appear in the pass bands of the second to fourth acoustic wave filters 4 to 6. Preferably, two higher-order modes among the first higher-order mode, the second higher-order mode, and the third higher-order mode do not appear in the pass bands of the second to fourth acoustic wave filters 4 to 6. For example, it is preferable that the responses of the first higher-order mode and the second higher-order mode, the responses of the first higher-order mode and the third higher-order mode, or the responses of the second higher-order mode and the third higher-order mode do not appear in the pass bands of the second to fourth acoustic wave filters 4 to 6. More preferably, all of the first higher-order mode, the second higher-order mode, and the third higher-order mode do not appear in the pass bands of the second to fourth acoustic wave filters 4 to 6. However, FIG. 5 is an example, and a positional relationship between the frequencies of the respective higher-order modes may be switched depending on conditions, such as the wavelength normalized thickness of the IDT electrode or the like, for example.

A feature of the multiplexer 1 according to the present preferred embodiment is that in at least one acoustic wave resonator of the first acoustic wave filter 3, the response of the first higher-order mode does not appear in the second to fourth pass bands illustrated in FIG. 4. Therefore, the filter characteristics of the second to fourth acoustic wave filters 4 to 6 do not easily deteriorate.

The features of the present preferred embodiment are described in the following i) and ii).

i) Values of a wavelength normalized thickness T_(LT) of the piezoelectric body 15 made of lithium tantalate, an Euler angle θ_(LT) of the piezoelectric body 15 made of lithium tantalate, a wavelength normalized thickness T_(S) of the silicon oxide film 14, a wavelength normalized thickness T_(N) of the silicon nitride film 13, a wavelength normalized thickness T_(E) of the IDT electrode 16 converted to a thickness of aluminum, a propagation orientation Nisi of the support substrate 12 made of silicon, and a wavelength normalized thickness T_(Si) of the support substrate 12 are set to values such that a frequency f_(hl_t) ^((n)) of the first higher-order mode determined by the following expression (1) and expression (2) satisfies the following expression (3) or expression (4) for all m which satisfy m>n, and ii) T_(Si)> about 20 is satisfied. Note that when the IDT electrode is defined by a laminated metal film, the wavelength normalized thickness T_(E) is a wavelength normalized thickness converted to the thickness of the IDT electrode made of aluminum, from a thickness and density of each electrode layer of the IDT electrode.

With this, the response by the first higher-order mode is located outside the pass band of the second to fourth acoustic wave filters 4 to 6. Accordingly, the filter characteristics of the second to fourth acoustic wave filters 4 to 6 do not easily deteriorate by the first higher-order mode. The fact that the frequency of the first higher-order mode is located outside the second to fourth pass bands by satisfying the above conditions will be described in more detail below.

$\begin{matrix} {V_{\text{?}} = {{a_{T_{\text{?}}}^{(3)}\left( {\left( {T_{\text{?}} + c_{\text{?}}} \right)^{3} + b_{\text{?}}^{(3)}} \right)} + {{a_{T_{\text{?}}}^{(2)}\left( {\left( {T_{\text{?}} + c_{\text{?}}} \right)^{2} + b_{\text{?}}^{(2)}} \right)}{a_{\text{?}}^{(1)}\left( {T_{\text{?}} + c_{\text{?}}} \right)}} + {a_{\text{?}}^{(\text{?})}\left( {\left( {T_{S} + c_{\text{?}}} \right)^{2} + b_{\text{?}}^{(2)}} \right)} + {a_{T_{S}}^{(1)}\left( {T_{S} + c_{T_{\text{?}}}} \right)} + {a_{\text{?}}^{(\text{?})}\left( {\left( {T_{N} + c_{\text{?}}} \right)^{2} + b_{\text{?}}^{(2)}} \right)} + {a_{T_{\text{?}}}^{(1)}\left( {T_{N} + c_{T_{\text{?}}}} \right)} + {a_{T_{E}}^{(1)}\left( {T_{E} + c_{T_{E}}} \right)} + {a_{\psi_{\text{?}}}^{(5)}\left( {\left( {\psi_{Si} + c_{\psi_{\text{?}}}} \right)^{5} + b_{\text{?}}^{(5)}} \right)} + {a_{\psi_{\text{?}}}^{(4)}\left( {\left( {\psi_{Si} + c_{\psi_{Si}}} \right)^{4} + b_{\psi_{\text{?}}}^{(4)}} \right)} + {a_{\psi_{Si}}^{(3)}\left( {\left( {\psi_{Si} + c_{\psi_{Si}}} \right)^{3} + b_{\psi_{Si}}^{(3)}} \right)} + {a_{\psi_{Si}}^{(2)}\left( {\left( {\psi_{Si} + c_{\psi_{\text{?}}}} \right)^{2} + b_{\psi_{Si}}^{(2)}} \right)} + {a_{\psi_{Si}}^{(1)}\left( {\psi_{Si} + c_{\psi_{Si}}} \right)} + e}} & {{Expression}\mspace{14mu} (1)} \\ {\mspace{79mu} {{f_{h_{s}\_ \; t}^{(n)} = \frac{V_{h_{s}\_ \; t}}{\lambda_{t}^{(n)}}},\left( {{s = 1},2,3} \right)}} & {{Expression}\mspace{14mu} (2)} \\ {\mspace{79mu} {f_{{hs}\; \_ \; t}^{(n)} > f_{u}^{(m)}}} & {{Expression}\mspace{14mu} (3)} \\ {\mspace{79mu} {{f_{{hs}\; \_ \; t}^{(n)} < f_{l}^{(m)}}{\text{?}\text{indicates text missing or illegible when filed}}}} & {{Expression}\mspace{14mu} (4)} \end{matrix}$

In place of an acoustic velocity V_(h) represented by the expression (1), it is more preferable to use the acoustic velocity V_(h) represented by the following expression (5). In this case, a ripple by the higher-order mode is more unlikely to occur in the other acoustic wave filter.

$\begin{matrix} {{V_{h} = {{a_{T_{LT}}^{(2)}\left( {\left( {T_{LT} - c_{T_{\text{?}}}} \right)^{2} - b_{T_{\text{?}}}^{(\text{?})}} \right)} + {a_{\text{?}}^{(1)}\left( {T_{LT} - c_{T_{\text{?}}}} \right)} + {a_{\text{?}}^{(2)}\left( {\left( {T_{S} - c_{T_{S}}} \right)^{2} - b_{T_{\text{?}}}^{(2)}} \right)} + {a_{T_{S}}^{(1)}\left( {T_{S} - c_{T_{\text{?}}}} \right)} + {a_{T_{N}}^{(3)}\left( {\left( {T_{N} - c_{T_{N}}} \right)^{3} - b_{T_{N}}^{(3)}} \right)} + {a_{T_{N}}^{(2)}\left( {\left( {T_{N} - c_{T_{N}}} \right)^{2} - b_{T_{N}}^{(2)}} \right)} + {a_{T_{N}}^{(1)}\left( {T_{N} - c_{T_{N}}} \right)} + {a_{T_{E}}^{(1)}\left( {T_{E} - c_{T_{E}}} \right)} + {a_{\psi_{Si}}^{(4)}\left( {\left( {\psi_{Si} - c_{\psi_{Si}}} \right)^{4} - b_{\psi_{Si}}^{(4)}} \right)} + {a_{\psi_{Si}}^{(3)}\left( {\left( {\psi_{Si} - c_{\psi_{Si}}} \right)^{3} - b_{\psi_{Si}}^{(3)}} \right)} + {a_{\psi_{Si}}^{(2)}\left( {\left( {\psi_{Si} - c_{\psi_{\text{?}}}} \right)^{2} - b_{\psi_{\text{?}}}^{(2)}} \right)} + {a_{\psi_{\text{?}}}^{(1)}\left( {\psi_{Si} - c_{\psi_{Si}}} \right)} + {a_{\theta_{LT}}^{(1)}\left( {\theta_{LT} - c_{\theta_{L\text{?}}}} \right)} + {{d_{T_{LT}T_{3}}\left( {T_{LT} - c_{T_{LT}}} \right)}\left( {T_{S} - {c_{T}}_{\text{?}}} \right)} + {{d_{\text{?}}\left( {T_{LT} - c_{LT}} \right)}\left( {T_{N} - c_{T_{N}}} \right)} + {{d_{\text{?}}\left( {T_{LT} - c_{T_{LT}}} \right)}\left( {\psi_{Si} - c_{\psi_{Si}}} \right)} + {{d_{T\text{?}}\left( {T_{S} - c_{T_{S}}} \right)}\left( {T_{N} - c_{T_{N}}} \right)} + {{d_{T\text{?}}\left( {T_{N} - c_{T_{N}}} \right)}\left( {\psi_{Si} - c_{\psi_{Si}}} \right)} + {{d_{T_{\text{?}}}\left( {T_{N} - c_{T_{N}}} \right)}\left( {\theta_{LT} - c_{\theta_{LT}}} \right)} + {{d_{T_{\text{?}}}\left( {T_{E} - c_{T_{E}}} \right)}\left( {\psi_{Si} - c_{\psi_{Si}}} \right)} + {{d_{\text{?}}\left( {\psi_{Si} - c_{\psi_{Si}}} \right)}\left( {\theta_{LT} - c_{\theta_{LT}}} \right)} + e}}{\text{?}\text{indicates text missing or illegible when filed}}} & {{Expression}\mspace{14mu} (5)} \end{matrix}$

Note that in the expression (1) to expression (4), h indicates a higher-order mode, m indicates an m-th (m>n) acoustic wave filter (m), and n indicates an n-th acoustic wave filter (n). Furthermore, t indicates a t-th element (acoustic wave resonator) in the n-th filter. Furthermore, f_(hs_t) ^((n)) indicates a frequency of a higher-order mode in the acoustic wave resonator (t) included in the acoustic wave filter (n). Note that s is 1, 2, or 3, f_(hs_t) ^((n)) indicates a frequency of the first higher-order mode when s is 1, indicates a frequency of the second higher-order mode when s is 2, and indicates a frequency of the third higher-order mode when s is 3. Furthermore, f_(u) ^((m)) is a frequency of a high band side end portion of the pass band in the acoustic wave filter (m). Furthermore, f₁ ^((m)) is a frequency of a low band side end portion of the pass band in the acoustic wave filter (m). In addition, in the present specification, the wavelength normalized thickness is a thickness obtained by normalizing a thickness by the wavelength of the IDT electrode. Here, the wavelength refers to a wavelength λ determined by an electrode finger pitch of the IDT electrode. Accordingly, the wavelength normalized thickness is a thickness obtained by normalizing an actual thickness with λ being 1, and is a value obtained by dividing the actual thickness by λ. Note that the wavelength λ determined by the electrode finger pitch of the IDT electrode may be determined by an average value of the electrode finger pitch. Here, λ_(t) ^((n)) is a wavelength determined by the electrode finger pitch of the IDT electrode in the acoustic wave resonator (t) included in the acoustic wave filter (n). In this specification, the wavelength normalized thickness may be simply described as a film thickness.

The inventors of preferred embodiments of the present application have discovered that the frequency position of the first higher-order mode is affected by the parameters described above.

As illustrated in FIG. 6, according to the propagation orientation ψ_(Si) of the support substrate made of silicon, the acoustic velocity in the main mode hardly changes but the acoustic velocity in the first higher-order mode greatly changes. As illustrated in FIG. 7, the acoustic velocity in the first higher-order mode changes according to the wavelength normalized thickness T_(LT) of the piezoelectric body made of lithium tantalate. As illustrated in FIG. 8, the acoustic velocity in the first higher-order mode also changes according to a cut-angle of the piezoelectric body made of lithium tantalate, that is, (about 90°+θ_(LT)). As illustrated in FIG. 9, the acoustic velocity in the first higher-order mode also slightly changes according to the wavelength normalized thickness T_(S) of the silicon oxide film. As illustrated in FIG. 10, it can be seen that the acoustic velocity in the first higher-order mode also changes according to the wavelength normalized thickness T_(N) of the silicon nitride film. Additionally, as illustrated in FIG. 11, the acoustic velocity in the first higher-order mode also slightly changes according to the wavelength normalized thickness T_(E) of the IDT electrode. The inventors of preferred embodiments of the present application freely changed these parameters to determine the acoustic velocity in the first higher-order mode. As a result, it has been discovered that the acoustic velocity in the first higher-order mode is expressed by the expression (1). Additionally, it has been confirmed that the coefficients in the expression (1) may be values shown in Table 7 indicated below for each crystal orientation of the support substrate made of silicon. Furthermore, it has been confirmed that the coefficients in the expression (5) may be values shown in Table 8 indicated below for each crystal orientation of the support substrate made of silicon.

TABLE 7 s = 1, First Higher- Order Mode Si(100) Si(110) Si(111) a_(TLT) ⁽³⁾ 0 0 0 a_(TLT) ⁽²⁾ 0 0 0 a_(TLT) ⁽¹⁾ −128.109974 −84.392576 −78.4352 b_(TLT) ⁽³⁾ 0 0 0 b_(TLT) ⁽²⁾ 0 0 0 c_(TLT) −0.2492038 −0.247604 −0.24838 a_(TS) ⁽²⁾ 0 0 0 a_(TS) ⁽¹⁾ −109.6889 −182.2936559 −485.867 b_(TS) ⁽²⁾ 0 0 0 c_(TS) −0.249363 −0.2498958 −0.24942 a_(TN) ⁽²⁾ −337.59528 −198.4171235 −264.804 a_(TN) ⁽¹⁾ −109.08389 38.137636 −20.3216 b_(TN) ⁽²⁾ −0.0262274 −0.04671597 −0.04389 c_(TN) −0.29617834 0.369166 −0.34988 a_(TE) ⁽¹⁾ 175.4682 13.0363945 0 c_(TE) −0.14826 −0.14979166 0 a_(ψSi) ⁽⁵⁾ 0 0 0 a_(ψSi) ⁽⁴⁾ 0 0.000489723 0.000503 a_(ψSi) ⁽³⁾ 0.0236358 −5.09E−05 0.006871 a_(ψSi) ⁽²⁾ −0.0357088 −1.017335189 −0.80395 a_(ψSi) ⁽¹⁾ −34.8157175 0 −5.57553 b_(ψSi) ⁽⁵⁾ 0 0 0 b_(ψSi) ⁽⁴⁾ 0 −2150682.513 −352545 b_(ψSi) ⁽³⁾ 0 −21460.18941 2095.948 b_(ψSi) ⁽²⁾ −288.415605 −970.8815104 −470.617 c_(ψSi) −22.5 −36.8125 −33.3025 e 5251.687898 5092.365583 4851.236

TABLE 8 s = 1, First Higher- Order Mode Si(100) Si(110) Si(111) a_(TLT) ⁽²⁾ 0 0 0 a_(TLT) ⁽¹⁾ 0 0 0 b_(TLT) ⁽²⁾ 0 0 0 c_(TLT) 0 0 0 a_(TS) ⁽²⁾ 0 0 0 a_(TS) ⁽¹⁾ 0 0 534.5188318 b_(TS) ⁽²⁾ 0 0 0 c_(TS) 0 0 0.249010293 a_(TN) ⁽³⁾ 0 0 0 a_(TN) ⁽²⁾ 0 0 0 a_(TN) ⁽¹⁾ 0 0 −36.51741324 b_(TN) ⁽³⁾ 0 0 0 b_(TN) ⁽²⁾ 0 0 0 c_(TN) 0 0 0.35114806 a_(TE) ⁽¹⁾ 0 0 0 c_(TE) 0 0 0 a_(ψSi) ⁽⁴⁾ 0 0 0.000484609 a_(ψSi) ⁽³⁾ 0.022075968 0 0.005818261 a_(ψSi) ⁽²⁾ −0.18782287 0.081701713 −0.805302317 a_(ψSi) ⁽¹⁾ −33.85785847 10.57201342 −4.785681077 b_(ψSi) ⁽⁴⁾ 0 0 351437.8188 b_(ψSi) ⁽³⁾ 806.2400011 0 −1862.605341 b_(ψSi) ⁽²⁾ 270.2635345 986.4812738 471.945355 c_(ψSi) 20.26171875 37.73795535 32.87410926 a_(θLT) ⁽¹⁾ 0 0 0 c_(θLT) 0 0 0 d_(TLTTS) 0 0 0 d_(TLTTN) 0 0 0 d_(TLTψSi) 0 0 0 d_(TSTN) 0 0 1862.994192 d_(TNψSi) 0 0 0 d_(TNθLT) 0 0 0 d_(TEψSi) 0 0 0 d_(ψSiθLT) 0 0 0 e 5317.859375 5103.813161 4853.204861

When the acoustic velocity of the first higher-order mode is denoted as V_(hl_t), the frequency of the first higher-order mode is represented by f_(hl_t) ^((n))=V_(hl_t)/λ_(t) ^((n)) based on the expression (2). Here, f_(hl) is the frequency of the first higher-order mode, and t is a number of an element such as the resonator or the like of the n-th filter.

In the present preferred embodiment, as indicated by the expression (3) or the expression (4), f_(hl_t) is higher than f_(u) ^((m)) or lower than f₁ ^((m)). That is, f_(hl_t) is lower than that of each of the low band side end portions or higher than that of each of the high band side end portions, of the second pass band, the third pass band, and the fourth pass band illustrated in FIG. 4. Therefore, it can be seen that the frequency f_(hl_t) ^((n)) of the first higher-order mode is not located within the second to fourth pass bands.

Here, as illustrated in FIG. 33, an Si (100) indicates a substrate obtained by cutting at a (100) plane orthogonal or substantially orthogonal to a crystal axis represented by a miller index [100] in a crystal structure of silicon having a diamond structure. It should be noted that a crystallographically equivalent plane such as an Si (010) or the like is also included.

As illustrated in FIG. 34, an Si (110) indicates a substrate obtained by cutting at a (110) plane orthogonal or substantially orthogonal to a crystal axis represented by a miller index [110] in a crystal structure of silicon having a diamond structure. It should be noted that another crystallographically equivalent plane is also included.

As illustrated in FIG. 35, an Si (111) indicates a substrate obtained by cutting at a (111) plane orthogonal or substantially orthogonal to a crystal axis represented by a miller index [111] in a crystal structure of silicon having a diamond structure. It should be noted that another crystallographically equivalent plane is also included.

In the above-described expression (1):

a) When the Si (100) (Euler angles are assumed to be (φ_(Si)= about 0±5°, θ_(Si)= about 0±5°, ψ_(Si))) is used, a range of ψ_(Si) is set to about 0°≤ψ_(Si) about 45°. However, because of the symmetry of the crystal structure of the Si (100), ψ_(Si) and ψ_(Si)±(n×90°) are synonymous with each other (n=1, 2, 3 . . . ). In the same manner, ψ_(Si) and −ψ_(Si) are synonymous with each other.

b) When the Si (110) (Euler angles are assumed to be (φ_(Si)= about −45±5°, θ_(Si)= about −90±5°, ψ_(Si))) is used, a range of ψ_(Si) is set to about 0°≤ψ_(Si) about 90°. However, because of the symmetry of the crystal structure of the Si (110), ψ_(Si) and ψ_(Si)±(n×180°) are synonymous with each other (n=1, 2, 3 . . . ). In the same manner, ψ_(Si) and −ψ_(Si) are synonymous with each other.

c) When the Si (111) (Euler angles are assumed to be (φ_(Si)= about −45±5°, θ_(Si)= about −54.73561±5°, ψ_(Si))) is used, a range of ψ_(Si) is set to about 0°≤ψ_(Si) about 60°. However, because of the symmetry of the crystal structure of the Si (111), ψ_(Si) and ψ_(Si)±(n×120°) are synonymous with each other (n=1, 2, 3 . . . ). In the same manner, ψ_(Si) and −ψ_(Si) are synonymous with each other.

In addition, although a range of θ_(LT) is set to about −180°<θ_(LT)≤ about 0°, θ_(LT) and θ_(LT)+180° may be handled as being synonymous with each other.

Note that in this specification, in the Euler angles (within the range of about 0°±5°, θ, within the range of about 0°±15°), “within the range of about 0°±5°” means “within a range of about −5° or more and about +5° or less”, and “within the range of about 0°±15°” means “within the range of about −15° or more and about +15° or less”.

The wavelength normalized thickness T_(E) of the IDT electrode 16 is a thickness obtained by converting the wavelength normalized thickness of the electrode layer of the IDT electrode 16 into the film thickness of the IDT electrode made of aluminum based on a density ratio. More specifically, the wavelength normalized thickness T_(E) is obtained by multiplying a value obtained by dividing the density of the IDT electrode 16 by the density of aluminum, and the wavelength normalized thickness of the IDT electrode 16. However, the electrode material is not limited to Al. Various metals, such as Ti, NiCr, Cu, Pt, Au, Mo, W, or the like, for example, can be used. Alternatively, an alloy including these metals as a main component may be used. Alternatively, a laminated metal film provided by laminating a plurality of metal films made of these metals or the alloy may be used. In this case, the wavelength normalized thickness T_(E) is a wavelength normalized thickness converted to the thickness of the IDT electrode made of aluminum, from the thickness and density of each electrode layer of the IDT electrode.

FIG. 12A is a diagram illustrating filter characteristics of a multiplexer of a comparative example in which the acoustic wave resonator does not satisfy the expression (3) or the expression (4), and FIG. 12B is a diagram illustrating filter characteristics of the multiplexer according to the first preferred embodiment.

FIGS. 12A and 12B each illustrate the filter characteristics of the first acoustic wave filter and the second acoustic wave filter. The solid line indicates the filter characteristics of the first acoustic wave filter. As indicated by the broken line in FIG. 12A, a ripple appears in the pass band in the filter characteristics of the second acoustic wave filter. This ripple is caused by a response of the higher-order mode of the acoustic wave resonator in the first acoustic wave filter. In contrast, as illustrated in FIG. 12B, in the multiplexer according to the first preferred embodiment, such a ripple does not appear in the pass band of the second acoustic wave filter. That is, since the acoustic wave resonator is configured so as to satisfy the expression (3) or the expression (4), the ripple does not appear in the second pass band of the second acoustic wave filter.

FIG. 13 is a diagram illustrating a relationship between the wavelength normalized thickness T_(Si) of the support substrate made of silicon and the phase maximum values of the first higher-order mode, the second higher-order mode, and the third higher-order mode. As is apparent from FIG. 13, it can be seen that when the wavelength normalized thickness T_(Si) of the support substrate made of silicon is larger than about 4λ, the magnitude of the response in the first higher-order mode becomes constant or substantially constant and sufficiently small. Note that when the wavelength normalized thickness T_(Si) of the support substrate is greater than about 10λ, the responses of the second and third higher-order modes also become small, and when the thickness is greater than about 20λ, those in all the first to third higher-order modes become sufficiently small. Therefore, preferably, the wavelength normalized thickness T_(Si) of the support substrate satisfies T_(Si)> about 4, for example. More preferably, the wavelength normalized thickness T_(Si) of the support substrate satisfies T_(Si)> about 10, for example. Still more preferably, the wavelength normalized thickness T_(Si) of the support substrate satisfies T_(Si)> about 20, for example.

In the present preferred embodiment, in at least one acoustic wave resonator of the plurality of acoustic wave resonators of the first acoustic wave filter 3, the frequency of the first higher-order mode satisfies the expression (3) or the expression (4). More preferably, in the acoustic wave resonator closest to the antenna terminal, the frequency position of the response in the higher-order mode satisfies the expression (3) or the expression (4). This is because the influence of a higher-order mode in the acoustic wave resonator closest to the antenna terminal tends to appear greatly in the pass bands of the second to fourth acoustic wave filters 4 to 6, in comparison with another acoustic wave resonator.

More preferably, in all of the acoustic wave resonators, the frequency position in the first higher-order mode satisfies the expression (3) or the expression (4). With this, the ripple caused by the response of the first higher-order mode is less likely to occur in the pass bands of the second to fourth acoustic wave filters 4 to 6.

When the structure of preferred embodiments of the present invention is applied, as described above, the higher-order mode tends to be confined in a portion where the silicon oxide film 14 and the piezoelectric body 15 are laminated, but by making the wavelength normalized thickness of the piezoelectric body 15 to be equal to or less than about 3.5λ, for example, the laminated portion of the silicon oxide film 14 and the piezoelectric body 15 becomes thinner, so that the higher-order mode is unlikely to be confined.

More preferably, the film thickness of the piezoelectric body 15 made of LiTaO₃ is equal to or less than about 2.5λ, for example, and in this case, an absolute value of a temperature coefficient of frequency TCF can be decreased. Additionally, preferably, the film thickness of the piezoelectric body 15 made of LiTaO₃ is equal to or less than about 1.5λ, for example. In this case, an electromechanical coupling coefficient can be easily adjusted. Additionally, more preferably, the film thickness of the piezoelectric body 15 made of LiTaO₃ is equal to or less than about 0.5λ, for example. In this case, the electromechanical coupling coefficient can be easily adjusted within a wide range.

Second Preferred Embodiment

In a second preferred embodiment of the present invention, a ripple in the second higher-order mode, not in the first higher-order mode, is not located in the pass bands of the second to fourth filters 4 to 6. This will be described with reference to FIG. 14 to FIG. 19.

As illustrated in FIG. 14, an acoustic velocity in the second higher-order mode changes according to the propagation orientation ψ_(Si). In the same or similar manner, as illustrated in FIG. 15, the acoustic velocity in the second higher-order mode also changes according to the wavelength normalized thickness T_(LT) of the piezoelectric body made of lithium tantalate. As illustrated in FIG. 16, the acoustic velocity in the second higher-order mode also changes according to a cut-angle (about 90°+θ_(LT)) of the piezoelectric body made of lithium tantalate. As illustrated in FIG. 17, the acoustic velocity in the second higher-order mode also changes according to the wavelength normalized thickness T_(S) of the silicon oxide film. As illustrated in FIG. 18, the acoustic velocity in the second higher-order mode also changes according to the wavelength normalized thickness T_(N) of the silicon nitride film. As illustrated in FIG. 19, the acoustic velocity in the second higher-order mode also changes according to the wavelength normalized thickness T_(E) of the IDT electrode. From the results illustrated in FIG. 14 to FIG. 19, in the same or similar manner as in the first preferred embodiment, it has been discovered that the acoustic velocity in the second higher-order mode is also represented by the expression (1) or the expression (5). However, in a case of the second higher-order mode, it is necessary for the coefficient in the expression (1) to have values shown in Table 9 indicated below for each crystal orientation of the support substrate made of silicon. Furthermore, in the case of the second higher-order mode, it is necessary for the coefficient in the expression (5) to have values shown in Table 10 indicated below for each crystal orientation of the support substrate made of silicon.

TABLE 9 s = 2, Second Higher- Order Mode Si(100) Si(110) Si(111) a_(TLT) ⁽³⁾ 0 0 0 a_(TLT) ⁽²⁾ 2285.602094 3496.38329 −2357.61 a_(TLT) ⁽¹⁾ −538.88053 −1081.86178 −1308.55 b_(TLT) ⁽³⁾ 0 0 0 b_(TLT) ⁽²⁾ −0.0016565 −0.001741462 −0.00166 c_(TLT) −0.251442 −0.2501547 −0.2497 a_(TS) ⁽²⁾ −3421.09725 −4927.3017 −3633.11 a_(TS) ⁽¹⁾ −1054.253 −992.33158 −1006.69 b_(TS) ⁽²⁾ −0.0016565 −0.2551083 −0.00166 c_(TS) −0.2514423 0.2551 −0.25019 a_(TN) ⁽²⁾ 1042.56084 −423.87007 −135.325 a_(TN) ⁽¹⁾ 159.11219 80.7948 −106.73 b_(TN) ⁽²⁾ −0.02613905 −0.05219411 −0.0486 c_(TN) −0.2961538 −0.36996 −0.39884 a_(TE) ⁽¹⁾ −171.153846 −637.391944 −585.696 c_(TE) 0.15 −0.151238 −0.14932 a_(ψSi) ⁽⁵⁾ 0 0 0 a_(ψSi) ⁽⁴⁾ 0 −0.00098215 −0.00016 a_(ψSi) ⁽³⁾ −0.0038938 −0.002109232 −0.00037 a_(ψSi) ⁽²⁾ −0.00306409 2.25463 0.224668 a_(ψSi) ⁽¹⁾ 2.8538478 23.6872515 1.243381 b_(ψSi) ⁽⁵⁾ 0 0 0 b_(ψSi) ⁽⁴⁾ 0 −2959279.229 −399785 b_(ψSi) ⁽³⁾ 234.60436 −21928.45828 5.712562 b_(ψSi) ⁽²⁾ −289.82063 −1407.041187 −535.077 c_(ψSi) 22.78846 −39.1640886 −29.9806 e 5282.98076 5338.606811 5411.395

TABLE 10 s = 2, Second Higher- Order Mode Si(100) Si(110) Si(111) a_(TLT) ⁽²⁾ 0 0 0 a_(TLT) ⁽¹⁾ −608.2898721 −1003.471473 −1270.018362 b_(TLT) ⁽²⁾ 0 0 0 c_(TLT) 0.25 0.253954306 0.249121666 a_(TS) ⁽²⁾ 0 0 0 a_(TS) ⁽¹⁾ −1140.654415 −1030.75867 −1039.830158 b_(TS) ⁽²⁾ 0 0 0 c_(TS) 0.249966079 0.255272408 0.250032531 a_(TN) ⁽³⁾ −3219.596725 0 2822.963403 a_(TN) ⁽²⁾ 555.8662451 0 −264.9680504 a_(TN) ⁽¹⁾ 465.8636149 53.04201209 −288.9461645 b_(TN) ⁽³⁾ 0.001081155 0 0.000392787 b_(TN) ⁽²⁾ 0.04654949 0 0.48934743 c_(TN) 0.378426052 0.376449912 0.392908263 a_(TE) ⁽¹⁾ 0 −622.7635558 −614.5885324 c_(TE) 0 0.151274165 0.156815224 a_(ψSi) ⁽⁴⁾ 0 −0.00096736 −0.000227305 a_(ψSi) ⁽³⁾ 0 −0.006772454 −0.000220017 a_(ψSi) ⁽²⁾ 0 2.203099663 0.31727324 a_(ψSi) ⁽¹⁾ 0.833288758 28.15768206 0.648998523 b_(ψSi) ⁽⁴⁾ 0 2959964.533 396965.3474 b_(ψSi) ⁽³⁾ 0 19143.61126 87.44425969 b_(ψSi) ⁽²⁾ 0 1447.367657 532.0008856 c_(ψSi) 22.51017639 40.50966608 29.90240729 a_(θLT) ⁽¹⁾ −1.501270796 −2.076046604 −2.376979261 c_(θLT) −52.08683853 −50.82249651 −49.04879639 d_(TLTTS) 0 0 0 d_(TLTTN) 0 0 0 d_(TLTψSi) 0 16.61849238 0 d_(TSTN) 0 1820.795615 1482.11565 d_(TNψSi) 0 3.625908485 −3.131543418 d_(TNθLT) 0 0 1607.412093 d_(TEψSi) 0 0 0 d_(ψSiθLT) 0 0 0.089566113 e 5326.187246 5356.1100930 5418.323508

From the acoustic velocity V_(h2_t) in the second higher-order mode obtained as described above, based on the expression (2), the frequency position f_(h2_t) ^((n))=V_(h2_t)/λ_(t) ^((n)) of the response in the second higher-order mode is obtained. In the second preferred embodiment, the frequency position f_(h2_t) ^((n)) of the second higher-order mode is set so as to satisfy the following expression (3A) or expression (4A). Accordingly, in the second preferred embodiment, the response of the second higher-order mode is located outside the second to fourth pass bands of the second to fourth acoustic wave filters 4 to 6. Accordingly, a ripple in the filter characteristics of the second to fourth acoustic wave filters 4 to 6 by the response in the second higher-order mode does not easily occur.

f _(h2_t) ^((n)) >f _(u) ^((m))  Expression (3A)

f _(h2_t) ^((n)) <f ₁ ^((m))  Expression (4A)

More preferably, in all of the acoustic wave resonators, the frequency position of the response in the second higher-order mode satisfies the expression (3A) or the expression (4A). With this, a ripple caused by the response of the second higher-order mode is less likely to occur in the pass bands of the second to fourth acoustic wave filters 4 to 6.

Third Preferred Embodiment

In a third preferred embodiment of the present invention, a ripple in the third higher-order mode, not that in the first higher-order mode, is not located in the pass bands of the second to fourth filters 4 to 6. This will be described with reference to FIG. 20 to FIG. 25.

As illustrated in FIG. 20, an acoustic velocity in the third higher-order mode changes according to the propagation orientation ψ_(Si). In the same or similar manner, as illustrated in FIG. 21, the acoustic velocity in the third higher-order mode also changes according to the wavelength normalized thickness T_(LT) of the piezoelectric body made of lithium tantalate. As illustrated in FIG. 22, the acoustic velocity in the third higher-order mode also changes according to a cut-angle (about 90°+θ_(LT)) of the piezoelectric body made of lithium tantalate. As illustrated in FIG. 23, the acoustic velocity in the third higher-order mode also changes according to the wavelength normalized thickness T_(S) of the silicon oxide film. As illustrated in FIG. 24, the acoustic velocity in the third higher-order mode also changes according to the wavelength normalized thickness T_(N) of the silicon nitride film. As illustrated in FIG. 25, the acoustic velocity in the third higher-order mode also changes according to the wavelength normalized thickness T_(E) of the IDT electrode. From the results illustrated in FIG. 20 to FIG. 25, in the same or similar manner as in the first preferred embodiment, it has been discovered that the acoustic velocity in the third higher-order mode is also represented by the expression (1) or the expression (5). However, in a case of the third higher-order mode, it is necessary for the coefficient in the expression (1) to have values shown in Table 11 indicated below for each crystal orientation of the support substrate made of silicon. Furthermore, in the case of the third higher-order mode, it is necessary for the coefficient in the expression (5) to have values shown in Table 12 indicated below for each crystal orientation of the support substrate made of silicon.

TABLE 11 s = 3, Third Higher- Order Mode Si(100) Si(110) Si(111) a_(TLT) ⁽³⁾ 0 0 0 a_(TLT) ⁽²⁾ 0 0 3595.754 a_(TLT) ⁽¹⁾ −782.3425 −1001.237815 −592.246 b_(TLT) ⁽³⁾ 0 0 0 b_(TLT) ⁽²⁾ 0 0 −0.00164 c_(TLT) −0.254819 −0.2578947 −0.25367 a_(TS) ⁽²⁾ −14897.59116 0 0 a_(TS) ⁽¹⁾ −599.8312 −686.9212563 −438.155 b_(TS) ⁽²⁾ −0.00162005 0 0 c_(TS) −0.25682 −0.25546558 −0.25562 a_(TN) ⁽²⁾ 0 0 0 a_(TN) ⁽¹⁾ 0 125.557557 15.72663 b_(TN) ⁽²⁾ 0 0 0 c_(TN) 0 −0.349392713 −0.40872 a_(TE) ⁽¹⁾ −154.8823 −764.8758717 −290.54 c_(TE) −0.14819277 −0.15303643 −0.15149 a_(ψSi) ⁽⁵⁾ 0 0 0 a_(ψSi) ⁽⁴⁾ 0 0 −0.00073 a_(ψSi) ⁽³⁾ 0.010467682 −0.000286554 −0.00318 a_(ψSi) ⁽²⁾ −0.196913569 0.67197739 0.969126 a_(ψSi) ⁽¹⁾ −0.3019959 0.197549 0.359421 b_(ψSi) ⁽⁵⁾ 0 0 0 b_(ψSi) ⁽⁴⁾ 0 −0.000204665 0 b_(ψSi) ⁽³⁾ 0 −14837.92017 670.2052 b_(ψSi) ⁽²⁾ −240.3687037 −1590.306348 −525.572 c_(ψSi) 24.4578313 −41.9028 −31.1239 e 5730.906036 5574.008097 5675.837

TABLE 12 s = 3, Third Higher- Order Mode Si(100) Si(110) Si(111) a_(TLT) ⁽²⁾ −14710.45271 0 0 a_(TLT) ⁽¹⁾ −764.4056124 −942.2882121 −582.1313356 b_(TLT) ⁽²⁾ 0.001558682 0 0 c_(TLT) 0.257243963 0.255649419 0.251712062 a_(TS) ⁽²⁾ −21048.18754 0 0 a_(TS) ⁽¹⁾ −508.6730943 −705.5211128 −400.0368899 b_(TS) ⁽²⁾ 0.001583662 0 0 c_(TS) 0.257243963 0.254751848 0.254367977 a_(TN) ⁽³⁾ 0 0 0 a_(TN) ⁽²⁾ 0 0 0 a_(TN) ⁽¹⁾ 0 97.59462013 24.94240828 b_(TN) ⁽³⁾ 0 0 0 b_(TN) ⁽²⁾ 0 0 0 c_(TN) 0 0.367793031 0.404280156 a_(TE) ⁽¹⁾ −276.7311066 −747.0884117 0 c_(TE) 0.1494796 0.152164731 0 a_(ψSi) ⁽⁴⁾ 0 0 −0.00075146 a_(ψSi) ⁽³⁾ 0.011363183 0.003532214 −0.002666357 a_(ψSi) ⁽²⁾ −0.23320473 0.218669312 1.006728665 a_(ψSi) ⁽¹⁾ 0.214067146 −11.24365221 0.523191515 b_(ψSi) ⁽⁴⁾ 0 0 381500.5075 b_(ψSi) ⁽³⁾ 180.0564368 20914.04622 −493.6094588 b_(ψSi) ⁽²⁾ 257.0498426 1548.182277 530.6814032 c_(ψSi) 22.31890092 39.72544879 30.82490272 a_(θLT) ⁽¹⁾ 0 0 −1.551626054 c_(θLT) 0 0 −49.16731518 d_(TLTTS) −13796.64706 0 1575.283126 d_(TLTTN) 0 0 0 d_(TLTψSi) 30.35701585 0 0 d_(TSTN) 0 0 0 d_(TNψSi) 0 0 0 d_(TNθLT) 0 0 0 d_(TEψSi) 28.27908094 0 0 d_(ψSiθLT) 0 0 −0.086544362 e 5700.075407 5563.854277 5688.418884

From the acoustic velocity V_(h3_t) in the third higher-order mode obtained as described above, based on the expression (2), the frequency position of the response of the third higher-order mode can be obtained by the frequency position f_(h3_t) ^((n))=V_(h3_t)/λ_(t) ^((n)) of the third higher-order mode. In the third preferred embodiment, the frequency position of the third higher-order mode is set so as to satisfy the following expression (3B) or expression (4B). Accordingly, in the third preferred embodiment, the response of the third higher-order mode is located outside the second to fourth pass bands of the second to fourth acoustic wave filters 4 to 6. Accordingly, a ripple in the filter characteristics of the second to fourth acoustic wave filters 4 to 6 by the response in the third higher-order mode does not easily occur.

f _(h3_t) ^((n)) >f _(u) ^((m))  Expression (3B)

f _(h3_t) ^((n)) <f _(l) ^((m))  Expression (4B)

More preferably, in all of the acoustic wave resonators, the frequency position of the response in the third higher-order mode satisfies the expression (3B) or the expression (4B). With this, a ripple caused by the response of the third higher-order mode is less likely to occur in the pass bands of the second to fourth acoustic wave filters 4 to 6.

Fourth Preferred Embodiment

A fourth preferred embodiment of the present invention satisfies all of the first preferred embodiment, the second preferred embodiment, and the third preferred embodiment. A specific structure of a multiplexer according to the fourth preferred embodiment is the same as or similar to those in the first to third preferred embodiments.

In the fourth preferred embodiment, when the acoustic velocities of the first, second, and third higher-order modes are denoted as V_(hl_t), V_(h2_t), and V_(h3_t), respectively, the frequency positions of the responses of the first to third higher-order modes indicated by the expression (2) are represented by f_(hs_t) ^((n))=V_(hs_t)/λ_(t) ^((n)). Here, s is 1, 2, or 3. In the fourth preferred embodiment, all of the frequency f_(hl_t) ^((n)) of the response in the first higher-order mode, the frequency f_(h2_t) ^((n)) of the response in the second higher-order mode, and the frequency f_(h3_t) ^((n)) of the response in the third higher-order mode are higher than f_(u) ^((m)) or lower than f₁ ^((m)). Accordingly, the responses of the first to third higher-order modes are located outside the second to fourth pass bands of the second to fourth acoustic wave filters 4 to 6. Accordingly, deterioration in the filter characteristics of the second to fourth acoustic wave filters is more unlikely to occur.

Therefore, when the conditions of the fourth preferred embodiment are summarized, f_(hs_t) ^((n)) (where, s is 1, 2, or 3) satisfies f_(hs_t) ^((n))>f_(u) ^((m)) or f_(hs_t) ^((n))<f_(l) ^((m)) in any case where s is 1, 2, or 3. Also in the fourth preferred embodiment, it is preferable that T_(Si)> about 20, for example, be satisfied, thus making it possible to decrease the magnitude itself of the response in each of the first to third higher-order modes.

In the fourth preferred embodiment, although the responses of the first higher-order mode, the second higher-order mode, and the third higher-order mode are not present in the pass bands of the second to fourth acoustic wave filters which are the other acoustic wave filters, two higher-order modes of the first to third higher-order modes, such as the first higher-order mode and the second higher-order mode, the first higher-order mode and the third higher-order mode, or the second higher-order mode and the third higher-order mode, may be located outside the pass bands of the second to fourth acoustic wave filters. In this case as well, the influence of the higher-order mode can be reduced to a greater extent than in the first to third preferred embodiments.

FIG. 26 is a diagram illustrating a relationship between a Q value and a film thickness of LiTaO₃ in an acoustic wave device in which, for example, a low acoustic velocity film made of an SiO₂ film having a film thickness of about 0.35λ and a piezoelectric body made of lithium tantalate having Euler angles (about 0°, about −40.0°, about 0°) are laminated on a high acoustic velocity support substrate made of silicon. The vertical axis in FIG. 26 represents a product of Q characteristics and a band width ratio (Δf) of the resonator. Note that the high acoustic velocity support substrate is a support substrate in which an acoustic velocity of a propagating bulk wave is higher than an acoustic velocity of the acoustic wave propagating through the piezoelectric body. The low acoustic velocity film is a film in which an acoustic velocity of a propagating bulk wave is lower than the acoustic velocity of the acoustic wave propagating through the piezoelectric body. Furthermore, FIG. 27 is a diagram illustrating a relationship between the film thickness of the LiTaO₃ film as the piezoelectric body and a temperature coefficient of frequency TCF. FIG. 28 is a diagram illustrating a relationship between the film thickness of the LiTaO₃ film and the acoustic velocity. As illustrated in FIG. 26, it is preferable that the film thickness of the LiTaO₃ film is equal to or less than about 3.5λ, for example. In this case, the Q value becomes higher than that in the case where the thickness exceeds about 3.5λ. More preferably, in order to further increase the Q value, the film thickness of the LiTaO₃ film is equal to or less than about 2.5λ, for example.

Furthermore, as illustrated in FIG. 27, when the film thickness of the LiTaO₃ film is equal to or less than about 2.5λ, an absolute value of the temperature coefficient of frequency TCF can be made smaller than that in a case where the film thickness exceeds about 2.5λ. More preferably, the film thickness of the LiTaO₃ film is equal to or less than about 2λ, for example, and in this case, the absolute value of the temperature coefficient of frequency TCF may be made equal to or less than about 10 ppm/° C., for example. In order to decrease the absolute value of the temperature coefficient of frequency TCF, it is more preferable to set the film thickness of the LiTaO₃ film to be equal to or less than about 1.5λ, for example.

As illustrated in FIG. 28, when the film thickness of the LiTaO₃ film exceeds about 1.5λ, a change in the acoustic velocity is extremely small.

However, as illustrated in FIG. 29, when the film thickness of the LiTaO₃ film is in a range of equal to or greater than about 0.05λ and equal to or less than about 0.5λ, for example, the band width ratio greatly changes. Accordingly, the electromechanical coupling coefficient can be adjusted in a wider range. Therefore, in order to widen the adjustment range of the electromechanical coupling coefficient and the band width ratio, it is preferable that the film thickness of the LiTaO₃ film is in a range of equal to or greater than about 0.05λ and equal to or less than about 0.5λ, for example.

FIG. 30 and FIG. 31 are diagrams illustrating relationships between the film thickness (λ) of the SiO₂ film and the acoustic velocity and between the film thickness and the electromechanical coupling coefficient, respectively. Here, a silicon nitride film, an aluminum oxide film, and a diamond were each used as a high acoustic velocity film below the low acoustic velocity film made of SiO₂. The high acoustic velocity film is a film in which the acoustic velocity of the propagating bulk wave is higher than the acoustic velocity of the acoustic wave propagating through the piezoelectric body. The film thickness of the high acoustic velocity film was set to about 1.5λ, for example. The acoustic velocity of the bulk wave in silicon nitride is about 6000 m/s, the acoustic velocity of the bulk wave in aluminum oxide is about 6000 m/s, and the acoustic velocity of the bulk wave in diamond is about 12800 m/s. As illustrated in FIG. 30 and FIG. 31, even when the material of the high acoustic velocity film and the film thickness of the SiO₂ film are changed, the electromechanical coupling coefficient and the acoustic velocity are hardly changed. In particular, as illustrated in FIG. 31, when the film thickness of the SiO₂ film is equal to or greater than about 0.1λ and equal to or less than about 0.5λ, for example, the electromechanical coupling coefficient hardly changes regardless of the material of the high acoustic velocity film. Additionally, as illustrated in FIG. 30, it can be seen that, when the film thickness of the SiO₂ film is equal to or greater than about 0.3λ and equal to or less than about 2λ, for example, the acoustic velocity hardly changes regardless of the material of the high acoustic velocity film. Accordingly, preferably, the film thickness of the low acoustic velocity film made of silicon oxide is equal to or less than about 2λ, and more preferably, is equal to or less than about 0.5λ, for example.

The acoustic wave device in each of the above preferred embodiments can be used as a component, such as a duplexer of a high-frequency front end circuit. An example of such a high-frequency front end circuit will be described below.

FIG. 32 is a schematic configuration diagram of a communication device including a high-frequency front end circuit. A communication device 240 includes an antenna 202, a high-frequency front end circuit 230, and an RF signal processing circuit 203. The high-frequency front end circuit 230 is connected to the antenna 202. The high-frequency front end circuit 230 includes a multiplexer 210 and amplifiers 221 to 224 as a power amplifier. The multiplexer 210 includes first to fourth filters 211 to 214. As the multiplexer 210, the multiplexer according to a preferred embodiment of the present invention described above can be used. The multiplexer 210 includes an antenna common terminal 225 connected to the antenna 202. One ends of the first to third filters 211 to 213 defining reception filters and one end of the fourth filter 214 defining a transmission filter are connected to the antenna common terminal 225 in common. Output ends of the first to third filters 211 to 213 are connected to the amplifiers 221 to 223, respectively. Furthermore, the amplifier 224 is connected to an input end of the fourth filter 214.

Output ends of the amplifiers 221 to 223 are connected to the RF signal processing circuit 203. An input end of the amplifier 224 is connected to the RF signal processing circuit 203.

A multiplexer according to a preferred embodiment of the present invention can be suitably used as the multiplexer 210 in the communication device 240 described above.

Note that the multiplexers according to preferred embodiments of the present invention may include only a plurality of transmission filters, or may include a plurality of reception filters. Note that the multiplexers according to preferred embodiments of the present invention include n band pass filters, and n is 2 or more. Accordingly, a duplexer is also a multiplexer according to a preferred embodiment of the present invention.

Preferred embodiments of the present invention can be widely used for communication apparatuses such as a cellular phone, for example, as a multiplexer applicable to a filter and a multi-band system, a front end circuit, and a communication device.

While preferred embodiments of the present invention have been described above, it is to be understood that variations and modifications will be apparent to those skilled in the art without departing from the scope and spirit of the present invention. The scope of the present invention, therefore, is to be determined solely by the following claims. 

What is claimed is:
 1. A multiplexer comprising: N acoustic wave filters each including one end connected in common and having different pass bands, where, N is an integer equal to or greater than 2; wherein when the N acoustic wave filters include a first acoustic wave filter, a second acoustic wave filter, . . . , and an N-th acoustic wave filter in order from a side of a lower frequency of the pass band, at least one acoustic wave filter n, where 1≤n<N, among the N acoustic wave filters excluding an acoustic wave filter having the highest frequency of the pass band includes one or more acoustic wave resonators; a t-th acoustic wave resonator among the one or more acoustic wave resonators includes: a support substrate having Euler angles (φ_(Si), θ_(Si), ψ_(Si)) and made of silicon; a silicon nitride film on the support substrate; a silicon oxide film on the silicon nitride film; a piezoelectric body on the silicon oxide film, having Euler angles (φ_(LT) in a range of about 0°±5°, θ_(LT), ψ_(LT) in a range of about 0°±15°), and made of lithium tantalate; and an interdigital transducer (IDT) electrode on the piezoelectric body; when a wavelength determined by an electrode finger pitch of the IDT electrode is denoted as λ in the t-th acoustic wave resonator, a thickness normalized by the wavelength λ is denoted as a wavelength normalized thickness, when setting values of T_(LT) as a wavelength normalized thickness of the piezoelectric body, θ_(LT) as an Euler angle of the piezoelectric body, T_(S) as a wavelength normalized thickness of the silicon oxide film, T_(N) as a wavelength normalized thickness of the silicon nitride film, T_(E) as a wavelength normalized thickness of the IDT electrode converted to a thickness of aluminum, obtained by a product of a value obtained by dividing a density of the IDT electrode by a density of aluminum and a wavelength normalized thickness of the IDT electrode, ψ_(Si) as a propagation orientation in the support substrate, and T_(Si) as a wavelength normalized thickness of the support substrate, at least one of frequencies f_(hs_t) ^((n)) of first, second, and third higher-order modes, where, s is 1, 2, or 3, and a case in which s is 1 indicates a frequency of the first higher-order mode, a case in which s is 2 indicates a frequency of the second higher-order mode, and a case in which s is 3 indicates a frequency of the third higher-order mode, determined by an expression (1) and an expression (2) below determined by the T_(LT), the θ_(LT), the T_(S), the T_(N), the T_(E), the ψ_(Si), and the T_(Si), and all m-th acoustic wave filters, wherein n<m≤N, each having a pass band of a higher frequency than a frequency of a pass band of the n-th acoustic wave filter satisfy an expression (3) or an expression (4) below: $\begin{matrix} {{V_{\text{?}} = {{a_{T_{\text{?}}}^{(3)}\left( {\left( {T_{\text{?}} + c_{\text{?}}} \right)^{3} + b_{\text{?}}^{(3)}} \right)} + {{a_{T_{\text{?}}}^{(2)}\left( {\left( {T_{\text{?}} + c_{\text{?}}} \right)^{2} + b_{\text{?}}^{(2)}} \right)}{a_{\text{?}}^{(1)}\left( {T_{\text{?}} + c_{\text{?}}} \right)}} + {a_{\text{?}}^{(\text{?})}\left( {\left( {T_{S} + c_{\text{?}}} \right)^{2} + b_{\text{?}}^{(2)}} \right)} + {a_{T_{S}}^{(1)}\left( {T_{S} + c_{T_{\text{?}}}} \right)} + {a_{\text{?}}^{(\text{?})}\left( {\left( {T_{N} + c_{\text{?}}} \right)^{2} + b_{\text{?}}^{(2)}} \right)} + {a_{T_{\text{?}}}^{(1)}\left( {T_{N} + c_{T_{\text{?}}}} \right)} + {a_{T_{E}}^{(1)}\left( {T_{E} + c_{T_{E}}} \right)} + {a_{\psi_{\text{?}}}^{(5)}\left( {\left( {\psi_{Si} + c_{\psi_{\text{?}}}} \right)^{5} + b_{\text{?}}^{(5)}} \right)} + {a_{\psi_{\text{?}}}^{(4)}\left( {\left( {\psi_{Si} + c_{\psi_{Si}}} \right)^{4} + b_{\psi_{\text{?}}}^{(4)}} \right)} + {a_{\psi_{Si}}^{(3)}\left( {\left( {\psi_{Si} + c_{\psi_{Si}}} \right)^{3} + b_{\psi_{Si}}^{(3)}} \right)} + {a_{\psi_{Si}}^{(2)}\left( {\left( {\psi_{Si} + c_{\psi_{\text{?}}}} \right)^{2} + b_{\psi_{Si}}^{(2)}} \right)} + {a_{\psi_{Si}}^{(1)}\left( {\psi_{Si} + c_{\psi_{Si}}} \right)} + e}};} & {{Expression}\mspace{14mu} (1)} \\ {\mspace{79mu} {{f_{h_{s}\_ \; t}^{(n)} = \frac{V_{h_{s}\_ \; t}}{\lambda_{t}^{(n)}}},{\left( {{s = 1},2,3} \right);}}} & {{Expression}\mspace{14mu} (2)} \\ {\mspace{79mu} {{f_{{hs}\; \_ \; t}^{(n)} > f_{u}^{(m)}};}} & {{Expression}\mspace{14mu} (3)} \\ {\mspace{79mu} {{{f_{{hs}\; \_ \; t}^{(n)} < f_{l}^{(m)}};}{\text{?}\text{indicates text missing or illegible when filed}}}} & {{Expression}\mspace{14mu} (4)} \end{matrix}$ the f_(hs_t) ^((n)) represents a frequency of a higher-order mode corresponding to the s in the t-th acoustic wave resonator included in n-th the acoustic wave filter; the λ_(t) ^((n)) is a wavelength determined by the electrode finger pitch of the IDT electrode in the t-th acoustic wave resonator included in the n-th acoustic wave filter; the f_(u)(m) is a frequency of a high band side end portion of the pass band in each of the m-th acoustic wave filters; the f_(l) ^((m)) is a frequency of a low band side end portion of the pass band in each of the m-th acoustic wave filters; and each coefficient in the expression (1) is each value shown in Table 1, Table 2, or Table 3 indicated below for each value of the s and each crystal orientation of the support substrate: TABLE 1 s = 1, First Higher- Order Mode Si(100) Si(110) Si(111) a_(TLT) ⁽³⁾ 0 0 0 a_(TLT) ⁽²⁾ 0 0 0 a_(TLT) ⁽¹⁾ −128.109974 −84.392576 −78.4352 b_(TLT) ⁽³⁾ 0 0 0 b_(TLT) ⁽²⁾ 0 0 0 c_(TLT) −0.2492038 −0.247604 −0.24838 a_(TS) ⁽²⁾ 0 0 0 a_(TS) ⁽¹⁾ −109.6889 −182.2936559 −485.867 b_(TS) ⁽²⁾ 0 0 0 c_(TS) −0.249363 −0.2498958 −0.24942 a_(TN) ⁽²⁾ −337.59528 −198.4171235 −264.804 a_(TN) ⁽¹⁾ −109.08389 38.137636 −20.3216 b_(TN) ⁽²⁾ −0.0262274 −0.04671597 −0.04389 c_(TN) −0.29617834 0.369166 −0.34988 a_(TE) ⁽¹⁾ 175.4682 13.0363945 0 c_(TE) −0.14826 −0.14979166 0 a_(ψSi) ⁽⁵⁾ 0 0 0 a_(ψSi) ⁽⁴⁾ 0 0.000489723 0.000503 a_(ψSi) ⁽³⁾ 0.0236358 −5.09E−05 0.006871 a_(ψSi) ⁽²⁾ −0.0357088 −1.01733189 −0.80395 a_(ψSi) ⁽¹⁾ −34.8157175 0 −5.57553 b_(ψSi) ⁽⁵⁾ 0 0 0 b_(ψSi) ⁽⁴⁾ 0 −2150682.513 −352545 b_(ψSi) ⁽³⁾ 0 −21460.18941 2095.94 b_(ψSi) ⁽²⁾ −288.415605 −36.8125 −33.3025 c_(ψSi) −22.5 −36.8125 −33.3025 e 5251.687898 5092.365583 4851.236

TABLE 3 s = 2, Second Higher- Order Mode Si(100) Si(110) Si(111) a_(TLT) ⁽³⁾ 0 0 0 a_(TLT) ⁽²⁾ 2285.602094 3496.38329 −2357.61 a_(TLT) ⁽¹⁾ −538.88053 −1081.86178 −1308.55 b_(TLT) ⁽³⁾ 0 0 0 b_(TLT) ⁽²⁾ −0.0016565 −0.001741462 −0.00166 c_(TLT) −0.251442 −0.2501547 −0.2497 a_(TS) ⁽²⁾ −3421.09725 −4927.3017 −3633.11 a_(TS) ⁽¹⁾ −1054.253 −992.33158 −1006.69 b_(TS) ⁽²⁾ −0.0016565 −0.2551083 −0.00166 c_(TS) −0.2514423 0.2551 −0.25019 a_(TN) ⁽²⁾ 1042.56084 −423.87007 −135.325 a_(TN) ⁽¹⁾ 159.11219 80.7948 −106.73 b_(TN) ⁽²⁾ −0.02613905 −0.05219411 −0.0486 c_(TN) −0.2961538 −0.36996 −0.39884 a_(TE) ⁽¹⁾ −171.153846 −637.391944 −585.696 c_(TE) 0.15 −0.151238 −0.14932 a_(ψSi) ⁽⁵⁾ 0 0 0 a_(ψSi) ⁽⁴⁾ 0 −0.00098215 −0.00016 a_(ψSi) ⁽³⁾ −0.0038938 −0.002109232 −0.00037 a_(ψSi) ⁽²⁾ −0.00306409 2.25463 0.224668 a_(ψSi) ⁽¹⁾ 2.8538478 23.6872514 1.243381 b_(ψSi) ⁽⁵⁾ 0 0 0 b_(ψSi) ⁽⁴⁾ 0 −2959279.229 −399785 b_(ψSi) ⁽³⁾ 234.60436 −21928.45828 5.712562 b_(ψSi) ⁽²⁾ −289.82063 −1407.041187 −535.077 c_(ψSi) 22.78846 −39.1640886 −29.9806 e 5282.98076 5338.606811 5411.395

TABLE 3 s = 3, Third Higher- Order Mode Si(100) Si(110) Si(111) a_(TLT) ⁽³⁾ 0 0 0 a_(TLT) ⁽²⁾ 0 0 3595.754 a_(TLT) ⁽¹⁾ −782.3425 −1001.237815 −592.246 b_(TLT) ⁽³⁾ 0 0 0 b_(TLT) ⁽²⁾ 0 0 −0.00164 c_(TLT) −0.254819 −0.2578947 −0.25367 a_(TS) ⁽²⁾ −14897.59116 0 0 a_(TS) ⁽¹⁾ −599.8312 −686.9212563 −438.155 b_(TS) ⁽²⁾ −0.00162005 0 0 c_(TS) −0.25682 −0.25546558 −0.25562 a_(TN) ⁽²⁾ 0 0 0 a_(TN) ⁽¹⁾ 0 125.557557 15.72663 b_(TN) ⁽²⁾ 0 0 0 c_(TN) 0 −0.349392713 −0.40872 a_(TE) ⁽¹⁾ −154.8823 −764.8758717 −290.54 c_(TE) −0.14819277 −0.15303646 −0.15149 a_(ψSi) ⁽⁵⁾ 0 0 0 a_(ψSi) ⁽⁴⁾ 0 0 −0.00073 a_(ψSi) ⁽³⁾ 0.010467682 −0.000286554 −0.00318 a_(ψSi) ⁽²⁾ −0.196913569 0.67197739 0.969126 a_(ψSi) ⁽¹⁾ −0.3019959 0.197549 0.359421 b_(ψSi) ⁽⁵⁾ 0 0 0 b_(ψSi) ⁽⁴⁾ 0 −0.000204665 0 b_(ψSi) ⁽³⁾ 0 −14837.92017 670.2052 b_(ψSi) ⁽²⁾ −240.3687037 −1590.306348 −525.572 c_(ψSi) 24.4578313 −41.9028 −31.1239 e 5730.906036 5574.008097 5675.837


2. A multiplexer comprising: N acoustic wave filters each including one end connected in common and having different pass bands, where, N is an integer equal to or greater than 2; wherein when the N acoustic wave filters include a first acoustic wave filter, a second acoustic wave filter, and . . . , an N-th acoustic wave filter in order from a side of a lower frequency of the pass band, at least one n-th acoustic wave filter, wherein 1≤n<N, among the N acoustic wave filters excluding an acoustic wave filter having the highest frequency of the pass band includes one or more acoustic wave resonators; a t-th acoustic wave resonator among the one or more acoustic wave resonators includes: a support substrate having Euler angles (φ_(Si), θ_(Si), ψ_(Si)) and made of silicon; a silicon nitride film on the support substrate; a silicon oxide film on the silicon nitride film; a piezoelectric body on the silicon oxide film, having Euler angles (φ_(LT) in a range of about 0°±5°, θ_(LT), ψ_(LT) in a range of about 0°±15°), and made of lithium tantalate; and an interdigital transducer (IDT) electrode on the piezoelectric body; when a wavelength determined by an electrode finger pitch of the IDT electrode is denoted as λ in the t-th acoustic wave resonator, a thickness normalized by the wavelength λ is denoted as a wavelength normalized thickness, when setting values of T_(LT) as a wavelength normalized thickness of the piezoelectric body, θ_(LT) as an Euler angle of the piezoelectric body, T_(S) as a wavelength normalized thickness of the silicon oxide film, T_(N) as a wavelength normalized thickness of the silicon nitride film, T_(E) as a wavelength normalized thickness of the IDT electrode converted to a thickness of aluminum, obtained by a product of a value obtained by dividing a density of the IDT electrode by a density of aluminum and a wavelength normalized thickness of the IDT electrode, ψ_(Si) as a propagation orientation in the support substrate, and T_(Si) as a wavelength normalized thickness of the support substrate, at least one of frequencies f_(hs_t) ^((n)) of first, second, and third higher-order modes, where s is 1, 2, or 3, and a case in which s is 1 indicates a frequency of the first higher-order mode, a case in which s is 2 indicates a frequency of the second higher-order mode, and a case in which s is 3 indicates a frequency of the third higher-order mode, determined by an expression (5) and an expression (2) below determined by the T_(LT), the θ_(LT), the T_(S), the T_(N), the T_(E), the ψ_(Si), and the T_(Si), and all m-th acoustic wave filters (n<m≤N) each having a pass band of a higher frequency than a frequency of a pass band of the n-th acoustic wave filter satisfy an expression (3) or an expression (4) below: $\begin{matrix} {{V_{h} = {{a_{T_{LT}}^{(2)}\left( {\left( {T_{LT} - c_{T_{\text{?}}}} \right)^{2} - b_{T_{\text{?}}}^{(\text{?})}} \right)} + {a_{\text{?}}^{(1)}\left( {T_{LT} - c_{T_{\text{?}}}} \right)} + {a_{\text{?}}^{(2)}\left( {\left( {T_{S} - c_{T_{S}}} \right)^{2} - b_{T_{\text{?}}}^{(2)}} \right)} + {a_{T_{S}}^{(1)}\left( {T_{S} - c_{T_{\text{?}}}} \right)} + {a_{T_{N}}^{(3)}\left( {\left( {T_{N} - c_{T_{N}}} \right)^{3} - b_{T_{N}}^{(3)}} \right)} + {a_{T_{N}}^{(2)}\left( {\left( {T_{N} - c_{T_{N}}} \right)^{2} - b_{T_{N}}^{(2)}} \right)} + {a_{T_{N}}^{(1)}\left( {T_{N} - c_{T_{N}}} \right)} + {a_{T_{E}}^{(1)}\left( {T_{E} - c_{T_{E}}} \right)} + {a_{\psi_{Si}}^{(4)}\left( {\left( {\psi_{Si} - c_{\psi_{Si}}} \right)^{4} - b_{\psi_{Si}}^{(4)}} \right)} + {a_{\psi_{Si}}^{(3)}\left( {\left( {\psi_{Si} - c_{\psi_{Si}}} \right)^{3} - b_{\psi_{Si}}^{(3)}} \right)} + {a_{\psi_{Si}}^{(2)}\left( {\left( {\psi_{Si} - c_{\psi_{\text{?}}}} \right)^{2} - b_{\psi_{\text{?}}}^{(2)}} \right)} + {a_{\psi_{\text{?}}}^{(1)}\left( {\psi_{Si} - c_{\psi_{Si}}} \right)} + {a_{\theta_{LT}}^{(1)}\left( {\theta_{LT} - c_{\theta_{L\text{?}}}} \right)} + {{d_{T_{LT}T_{3}}\left( {T_{LT} - c_{T_{LT}}} \right)}\left( {T_{S} - {c_{T}}_{\text{?}}} \right)} + {{d_{\text{?}}\left( {T_{LT} - c_{LT}} \right)}\left( {T_{N} - c_{T_{N}}} \right)} + {{d_{\text{?}}\left( {T_{LT} - c_{T_{LT}}} \right)}\left( {\psi_{Si} - c_{\psi_{Si}}} \right)} + {{d_{T\text{?}}\left( {T_{S} - c_{T_{S}}} \right)}\left( {T_{N} - c_{T_{N}}} \right)} + {{d_{T\text{?}}\left( {T_{N} - c_{T_{N}}} \right)}\left( {\psi_{Si} - c_{\psi_{Si}}} \right)} + {{d_{T_{\text{?}}}\left( {T_{N} - c_{T_{N}}} \right)}\left( {\theta_{LT} - c_{\theta_{LT}}} \right)} + {{d_{T_{\text{?}}}\left( {T_{E} - c_{T_{E}}} \right)}\left( {\psi_{Si} - c_{\psi_{Si}}} \right)} + {{d_{\text{?}}\left( {\psi_{Si} - c_{\psi_{Si}}} \right)}\left( {\theta_{LT} - c_{\theta_{LT}}} \right)} + e}};} & {{Expression}\mspace{14mu} (5)} \\ {\mspace{79mu} {{f_{h_{s}\_ \; t}^{(n)} = \frac{V_{h_{s}\_ \; t}}{\lambda_{t}^{(n)}}},{\left( {{s = 1},2,3} \right);}}} & {{Expression}\mspace{14mu} (2)} \\ {\mspace{79mu} {{f_{{hs}\; \_ \; t}^{(n)} > f_{u}^{(m)}};}} & {{Expression}\mspace{14mu} (3)} \\ {\mspace{79mu} {{{f_{{hs}\; \_ \; t}^{(n)} < f_{l}^{(m)}};}{\text{?}\text{indicates text missing or illegible when filed}}}} & {{Expression}\mspace{14mu} (4)} \end{matrix}$ the f_(hs_t) ^((n)) represents a frequency of a higher-order mode corresponding to the s in the t-th acoustic wave resonator included in the n-th acoustic wave filter; the λ_(t) ^((n)) is a wavelength determined by the electrode finger pitch of the IDT electrode in the t-th acoustic wave resonator included in the n-th acoustic wave filter; the f_(u) ^((m)) is a frequency of a high band side end portion of the pass band in each of the m-th acoustic wave filters; the f_(l) ^((m)) is a frequency of a low band side end portion of the pass band in each of the m-th acoustic wave filters; and each coefficient in the expression (5) is each value shown in Table 4, Table 5, or Table 6 below for each value of the s and each crystal orientation of the support substrate: TABLE 4 s = 1, First Higher- Order Mode Si(100) Si(110) Si(111) a_(TLT) ⁽²⁾ 0 0 0 a_(TLT) ⁽¹⁾ 0 0 0 b_(TLT) ⁽²⁾ 0 0 0 c_(TLT) 0 0 0 a_(TS) ⁽²⁾ 0 0 0 a_(TS) ⁽¹⁾ 0 0 534.5188318 b_(TS) ⁽²⁾ 0 0 0 c_(TS) 0 0 0.249010293 a_(TN) ⁽³⁾ 0 0 0 a_(TN) ⁽²⁾ 0 0 0 a_(TN) ⁽¹⁾ 0 0 −36.51741324 b_(TN) ⁽³⁾ 0 0 0 b_(TN) ⁽²⁾ 0 0 0 c_(TN) 0 0 0.35114806 a_(TE) ⁽¹⁾ 0 0 0 c_(TE) 0 0 0 a_(ψSi) ⁽⁴⁾ 0 0 0.000484609 a_(ψSi) ⁽³⁾ 0.022075968 0 0.005818261 a_(ψSi) ⁽²⁾ −0.18782287 0.081701713 −0.805302371 a_(ψSi) ⁽¹⁾ −33.85785847 10.57201342 −4.785681077 b_(ψSi) ⁽⁴⁾ 0 0 351437.8188 b_(ψSi) ⁽³⁾ 806.2400011 0 −1862.605341 b_(ψSi) ⁽²⁾ 270.2635345 986.4812738 471.945355 c_(ψSi) 20.26171875 37.73795535 32.87410926 a_(θLT) ⁽¹⁾ 0 0 0 c_(θLT) 0 0 0 d_(TLTTS) 0 0 0 d_(TLTTN) 0 0 0 d_(TLTψSi) 0 0 0 d_(TSTN) 0 0 1862.994192 d_(TNψSi) 0 0 0 d_(TNθLT) 0 0 0 d_(TEψSi) 0 0 0 d_(ψSiθLT) 0 0 0 e 5317.859375 5103.813161 4853.204861

TABLE 5 s = 2, Second Higher- Order Mode Si(100) Si(110) Si(111) a_(TLT) ⁽²⁾ 0 0 0 a_(TLT) ⁽¹⁾ −608.2898721 −1003.471473 −1270.018362 b_(TLT) ⁽²⁾ 0 0 0 c_(TLT) 0.25 0.253954306 0.249121666 a_(TS) ⁽²⁾ 0 0 0 a_(TS) ⁽¹⁾ −1140.654415 −1030.75867 −1039.830158 b_(TS) ⁽²⁾ 0 0 0 c_(TS) 0.249966079 0.255272408 0.250032531 a_(TN) ⁽³⁾ −3219.596725 0 2822.963403 a_(TN) ⁽²⁾ 555.8662451 0 −264.9680504 a_(TN) ⁽¹⁾ 465.8636149 53.04201209 −288.9461645 b_(TN) ⁽³⁾ 0.001081155 0 0.000392787 b_(TN) ⁽²⁾ 0.04654949 0 0.48934743 c_(TN) 0.378426052 0.376449912 0.392908263 a_(TE) ⁽¹⁾ 0 −622.7635558 −614.5885324 c_(TE) 0 0.151274165 0.156815224 a_(ψSi) ⁽⁴⁾ 0 −0.00096736 −0.000227305 a_(ψSi) ⁽³⁾ 0 −0.006772454 −0.00220017 a_(ψSi) ⁽²⁾ 0 2.203099663 0.31727324 a_(ψSi) ⁽¹⁾ 0.833288758 28.15768206 0.648998523 b_(ψSi) ⁽⁴⁾ 0 2959964.533 396965.3474 b_(ψSi) ⁽³⁾ 0 19143.61126 87.44425969 b_(ψSi) ⁽²⁾ 0 1447.367657 532.0008856 c_(ψSi) 22.51017639 40.50966608 29.90240729 a_(θLT) ⁽¹⁾ −1.501270796 −2.076046604 −2.376979261 c_(θLT) −52.08683853 −50.82249561 −49.04879636 d_(TLTTS) 0 0 0 d_(TLTTN) 0 0 0 d_(TLTψSi) 0 16.61849238 0 d_(TSTN) 0 1820.795615 1482.11565 d_(TNψSi) 0 3.625908485 −3.131543418 d_(TNθLT) 0 0 1607.412093 d_(TEψSi) 0 0 0 d_(ψSiθLT) 0 0 0.089566113 e 5326.187246 5356.110093 5418.323508

TABLE 6 s = 3, Third Higher- Order Mode Si(100) Si(110) Si(111) a_(TLT) ⁽²⁾ −14710.45271 0 0 a_(TLT) ⁽¹⁾ −764.4056124 −942.2882121 −582.1313356 b_(TLT) ⁽²⁾ 0.001558682 0 0 c_(TLT) 0.257243963 0.255679719 0.251712062 a_(TS) ⁽²⁾ −21048.18754 0 0 a_(TS) ⁽¹⁾ −508.6730943 −705.5211128 −400.0368899 b_(TS) ⁽²⁾ 0.001583662 0 0 c_(TS) 0.257243963 0.254751848 0.254357977 a_(TN) ⁽³⁾ 0 0 0 a_(TN) ⁽²⁾ 0 0 0 a_(TN) ⁽¹⁾ 0 97.59462013 24.94240828 b_(TN) ⁽³⁾ 0 0 0 b_(TN) ⁽²⁾ 0 0 0 c_(TN) 0 0.367793031 0.404280156 a_(TE) ⁽¹⁾ −276.7311066 −747.0884117 0 c_(TE) 0.1494796 0.152164731 0 a_(ψSi) ⁽⁴⁾ 0 0 −0.00075146 a_(ψSi) ⁽³⁾ 0.011363183 0.003532214 −0.002666357 a_(ψSi) ⁽²⁾ −0.23320473 0.218669312 1.006728665 a_(ψSi) ⁽¹⁾ 0.214067146 −11.24365221 0.523191515 b_(ψSi) ⁽⁴⁾ 0 0 381500.5075 b_(ψSi) ⁽³⁾ 180.0564368 20914.04622 −493.6094588 b_(ψSi) ⁽²⁾ 257.0498426 1548.182277 530.6814032 c_(ψSi) 22.31890092 39.72544879 30.82490272 a_(θLT) ⁽¹⁾ 0 0 −1.551626054 c_(θLT) 0 0 −49.16731518 d_(TLTTS) −13796.64706 0 1575.283126 d_(TLTTN) 0 0 0 d_(TLTψSi) 30.35701585 0 0 d_(TSTN) 0 0 0 d_(TNψSi) 0 0 0 d_(TNθLT) 0 0 0 d_(TEψSi) 28.27908094 0 0 d_(ψSiθLT) 0 0 −0.086544362 e 5700.075407 5563.854277 5688.418884


3. The multiplexer according to claim 1, wherein the values of the T_(LT), the θ_(LT), the T_(S), the T_(N), the T_(E), the ψ_(Si), and the T_(Si), are selected such that the frequencies f_(hs_t) ^((n)) of the first and second higher-order modes satisfy the expression (3) or the expression (4).
 4. The multiplexer according to claim 1, wherein the values of the T_(LT), the θ_(LT), the T_(S), the T_(N), the T_(E), the ψ_(Si), and the T_(Si), are selected such that the frequencies f_(hs_t) ^((n)) of the first and third higher-order modes satisfy the expression (3) or the expression (4).
 5. The multiplexer according to claim 1, wherein the values of the T_(LT), the θ_(LT), the T_(S), the T_(N), the T_(E), the ψ_(Si), and the T_(Si), are selected such that the frequencies f_(hs_t) ^((n)) of the second and third higher-order modes satisfy the expression (3) or the expression (4).
 6. The multiplexer according to claim 1, wherein the values of the T_(LT), the θ_(LT), the T_(S), the T_(N), the T_(E), the ψ_(Si), and the T_(Si), are selected such that all the frequencies f_(hs_t) ^((n)) of the first, second, and third higher-order modes satisfy the expression (3) or the expression (4).
 7. The multiplexer according to claim 1, wherein the wavelength normalized thickness T_(Si) of the support substrate satisfies T_(Si)> about
 4. 8. The multiplexer according to claim 7, wherein T_(Si)> about 10 is satisfied.
 9. The multiplexer according to claim 8, wherein T_(Si)> about 20 is satisfied.
 10. The multiplexer according to claim 1, wherein the wavelength normalized thickness of the piezoelectric body is equal to or less than about 3.5λ.
 11. The multiplexer according to claim 10, wherein the wavelength normalized thickness of the piezoelectric body is equal to or less than about 2.5λ.
 12. The multiplexer according to claim 10, wherein the wavelength normalized thickness of the piezoelectric body is equal to or less than about 1.5λ.
 13. The multiplexer according to claim 10, wherein the wavelength normalized thickness of the piezoelectric body is equal to or less than about 0.5λ.
 14. The multiplexer according to claim 1, the multiplexer further comprising: an antenna terminal to which one ends of the plurality of acoustic wave filters are connected in common; wherein the acoustic wave resonator satisfying the expression (3) or the expression (4) is an acoustic wave resonator of the one or more acoustic wave resonators which is closest to the antenna terminal.
 15. The multiplexer according to claim 1, wherein all of the one or more acoustic wave resonators are each the acoustic wave resonator satisfying the expression (3) or the expression (4).
 16. The multiplexer according to claim 1, wherein the multiplexer is a duplexer.
 17. The multiplexer according to claim 1, further comprising: an antenna terminal to which one ends of the plurality of acoustic wave filters are connected in common; wherein the multiplexer is a composite filter in which three or more acoustic wave filters are connected in common on a side of the antenna terminal.
 18. The multiplexer according to claim 17, wherein the multiplexer is a composite filter device for carrier aggregation.
 19. The multiplexer according to claim 1, wherein the acoustic wave filter including the one or more acoustic wave resonators is a ladder filter including a plurality of serial arm resonators and a plurality of parallel arm resonators. 